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On the Parameters determining the Neutrino Flux from observed Active Galactic Nuclei

Extrapolating from a sample of 39 AGNs, we examine the impact on the total number of high energy muon-neutrino induced events (PeV-EeV) expected in IceCube (a 1Km^3 ``neutrino telescope''), due to variations in different parameters involved in the neutrino flux, such as the emission region geometry, the estimation models and distributions of the Doppler factor and the variability time. This work has been done taking into account different limits of the extragalactic neutrino flux. Among our conclusions, we find, in the case of the largest variability time, that the cylindrical geometry hypothesis for the emission region, produce a separation of 3 sigma in the total number of events relative to the spherical hypothesis. In addition, for similar choices of the burst time, spherical geometry and for the upper neutrino flux bound, we obtain a separation of 2.5 sigma in the total number of events, for some of the Doppler factor estimations. These differences are undistinguishable for other input values.

In particular, AGNs are very attractive to investigate, given that they are one of the most powerful continuous sources of the highest energy particles in the universe.
Among these particles, AGNs produce copiously neutrinos. Due to their weak interaction with matter, neutrinos keep their flux unaltered in their travel from the AGN to Earth. This feature makes them an ideal tool for exploring physical processes within the AGNs.
At present, our knowledge about the parameters involved in the calculation of the extragalactic neutrino flux is not so accurate. In fact, there is not an absolute agreement about some aspects related to these parameters. This is the case of the model used to estimate the Doppler boosting factor (e.g. Equipartition [9,10], Inverse Compton [9,10] and Variability [11] models) and its distribution (power-law-like), the shape of the emission region [12] (e.g. spherical and cylindrical) and the variability time scale, which can vary from few minutes to many days. A good determination of the Doppler factor is a key ingredient to establish the intrinsic photon luminosity, directly related with the neutrino flux. In the same way, the shape of the emission region changes the volume considered in the particle density of target photons that interact with protons. In addition, the time of burst is inversely proportional to the neutrino flux, so that, if the variability time decreases, the neutrino flux increases.
It is straightforward to note that changes in these parameters affect directly the neutrino flux and, hence, imply modifications in the expected value of the total number of neutrinos that will be detected on Earth. Consequently, it would be interesting to know the range of variation of these events when different assumptions are made about the parameters. If the number of events changes in a noticeable way, these numbers can be used as a tool to test the model parameters.
In this paper, we study the impact of these variations in the total number of muon neutrino induced events that could be detected in IceCube [13]. This detector is one of the so called "neutrino telescopes" (e.g. Amanda, Antares, Nestor) and consists of arrays of photomultipliers, located in a Cherenkov medium (ice), under construction in the South Pole with a standard volume of 1Km 3 . We choose ten years of exposure in the energy range of PeV-EeV. In spite of the fact, that this interval must contain a small number of events, it is convenient to be studied, considering the absence of background in this energy region. We model the neutrino flux based on a catalogue of 39 AGNs and extrapolate for a maximum flux given for different proposed limits, which are the Waxman-Bahcall [14], the benchmark used by the IceCube Collaboration [15] and the γ-ray bound [16]. The Unified Model also claims other characteristics of the AGNs, such as: two jets 1 moving in opposite directions and outwards to the accretion disk, an optically thick torus and radio lobes. Furthermore, for this model Seyferts, Quasars, Blazars and Radio Galaxies, are all AGN, only with different orientations as viewed from Earth, accretion rates and black hole masses. In this work, we assume this model including in our test sample all these types of galaxies.

III. NEUTRINO FLUX FROM AGN
We will consider as our working hypothesis [18] that within inner regions of AGN jets, protons are highly accelerated by sheet-like shock waves (blobs, knots or bursts) to energies up to the range of PeV-EeV, according to Fermi's stochastic acceleration. These protons should then interact with ambient photons emitted by the accretion disk from the thermal blackbody radiation (UV bump or blue bump, between 7 × 10 14 Hz to 3 × 10 15 Hz), leading to the ∆ + (1232) resonance and followed by the decays (pion from photo-production) nπ + → µ + ν µ → ν µ e + ν e . Thus, through this cascade, a neutrino flux is produced.
1 Generic AGN models do not necessarily involve jets.
In order to calculate the observed neutrino flux produced in each AGN, we begin with the following generic useful relation where the integration limits depend on the range of energies chosen to observe (in our case PeV-EeV), L ν is the observed neutrino luminosity and d L is the luminosity distance to the AGN, defined as with z the redshift of the AGN and d m the proper motion distance (see Eq. (B2) in App B).
The energy spectrum, dΦν dEν , will be assumed to be proportional to The typical power law energy dependence of the extragalactic neutrino flux is a resemblance of the proton spectrum, which is supposed to be accelerated by the Fermi's shock mechanism.
The slope of the power law, p, is usually taken to be 2, the generic E −2 ν . However, it can be considered in the range 1.3−2.7. The spectrum also includes an exponential cutoff depending on the AGNs maximum attainable energy, E AGN cut (we will assume a general cutoff of 10 10 GeV). Therefore, the exact neutrino spectrum formula is We focus our attention on the observable quantities that compose the neutrino luminosity.
Then, to unfold this luminosity in terms of these observables, we start from the definition of luminosity where N ν is the number of produced neutrinos, E ν the mean neutrino energy and ∆t obs the observed variability time of the emission region. Now, we give N ν from the neutrino production sequence explained at the beginning of the section where "br" stands for the branching ratio of the desired decay channel. This gives approxi- It is useful to express the neutrino energy in terms of the energy of the parent proton by where x represents the average fraction of energy transferred from the parent particle to the product, which numerically gives E ν ≈ 1 4 E µ ≈ 1 4 ( 1 5 E p ). On the other hand, to find the number of produced ∆'s we use with Φ p the initial flux of highly accelerated protons, N γ the number of ambient photons from the UV bump and σ pπ→∆ =10 −32 m 2 the delta photo-production cross section. Φ p and N γ are given by where L p and L γ are the proton and photon observed luminosities, respectively, E γ is the mean photon energy (in the present work we take it to be 10 eV), R is the observed size 2 of the emission region and V the observed volume of the emission region. It must be noted that the size of the emission region and, consequently, its volume depend on four parameters: the Doppler factor (δ), the variability time scale (∆t obs ), the redshift (z) and the angle between the velocity vector of the knot and the line of sight (θ). The detailed description of R and the volume can be found in Sec. III A 2.
Replacing Eqs. (6 -9) into Eq. (5), we obtain To write Eq. (10) in a more compact notation, we use the optical depth, τ , defined as where λ pγ→∆ is the proton interaction length, R, the observed size, which should be understood as the linear path traveled by the proton in the direction of movement through the ambient photons and we write K as where we have used br ∆→π = 0.5, br π→ν = 0.9998, x p→π = 0.2 and x π→ν = 1 4 . Since τ is a Lorentz invariant, we can go from the observer's frame to the frame attached to the knot, so that we can use a simple geometry (see Sec. III A 2) to describe the emission region, leaving τ as follows where primed quantities refer to a quantity measured in the jet and recalling that Lγ ∆t obs Eγ is itself a Lorentz invariant. This now gives for Eq. (10) However, not all L p will be observed because part of it would have already interacted with the ambient photons to produce the ∆'s. Therefore, the intrinsic luminosity will be e τ times the observed luminosity, L p = e τ L p obs . In our analysis we take L p obs , which is still unknown, as 10% of the photon total luminosity [18]. In addition, we must take into account the possible absorption of π + into the source when the medium is optically thick and therefore will not give time for the π + to decay into neutrinos. In this case, we use the substitution suggested by Ref. [18], changing τ by (1 − e −τ ). Thus, the factor (1 − e −τ )e (1−e −τ ) accounts for the absorption of p and π + in the source. Finally the ν µ and ν µ differential flux emitted

A. Flux parameters to be analyzed
In this subsection we describe the different hypothesis in the neutrino flux parameters studied in this paper.

Doppler boosting factor
As we have shown in the section before, the Doppler factor, δ, is a piece in the flux formula.
At present, there are various models for computing δ, which, in some cases, give different output values of δ for the same object. This disagreement could also imply differences in the flux that would be interesting to examine. For our work, we consider the following estimation models of δ: • Equipartition δ (EQ) [9,10]: This is calculated assuming the equipartition of energy between radiating particles and magnetic field and is obtained from the ratio of the observed and the maximum intrinsic brightness temperatures. The maximum intrinsic temperature is supposed to be the equipartition temperature.
• Inverse Compton δ (IC) [9,10]: In this case, the Doppler factor is determined from the comparison of the observed and predicted X-ray fluxes, assuming that this is caused by inverse Compton scattering of synchrotron photons off the radiating particles.
• Variability δ (Var) [11]: This is obtained in a similar way to δ EQ , but is weaker than this, since it takes the third root of temperatures' ratio. Additionally, it replaces the observed brightness temperature instead of the variability brightness temperature, which is estimated from a total flux density flares associated with VLBI components emerging from the AGN core, covering nearly 20 years of variations.
• Minimum δ (Min) [19]: This model is computed only from the proper motion data using the minimum δ value allowed by where Γ is the Lorentz factor and β app the apparent transverse velocity in units of c (the speed of light). We are going to use this model independent estimation as a reference.
Due to different proper motion observations of underlying shocks for a single jet, there should be a rather broad range of Doppler factors [20]. Then, it follows that a certain distribution in δ must be taken into account when computing the neutrino flux. According to Ref. [20] it is consistent with a steep power law distribution of intrinsic Doppler factors. In our calculation, we fit with a power law (δ −γ ) the data obtained for each type of calculation of δ, as described in Appendix. B. In Sec. V we explain how the neutrino flux is averaged over the fitted distribution.

Emission Region Geometry
In the study of the emission region geometry, we consider two possibilities for its shape in the proper frame: a spherical and a cylindrical volume. These models are based on the observed variability time ∆t obs [12]. However, we should state that ∆t obs could give, at most, the estimation of only one of the dimension of this region and therefore the other two should be inferred somehow from this. In this sense, the formulas given here are actually not relativistic transformations (as in App. A), but rather estimations and should not be misunderstood.
To understand the next formulas, it is appropriate to define a quantity D = δ 1+z that includes both the Doppler factor (δ) and the cosmological effects given by the redshift (z).
The geometries are described as follows: • Spherical geometry: it has two options for the jet-frame source radius R ′ . The first option, that we call SphereA, uses the common formula The second option, that we call by SphereB, uses a refined formula for R ′ , that consider the angle to the line of sight, θ, and the shock velocity, β shock (used value β shock =0.5). This expression is given by [12] R ′ = Dc∆t obs • Cylindrical geometry: with radius R ′ and jet-frame length l. The region is rapidly energized by a plane shock traveling along the jet, such that photons are emitted immediately after shock passage from a thin disk-like region downstream of the shock.
From here on, we assume l = aR ′ , with a variable (used value a = 2).

Variability Time Scale
The observed fluctuations over the whole spectrum in the AGNs, or variability time scales of the bursts, are one of the main characteristics of Active Galaxies [21]. This variability suggested that the emitting regions were extremely compact, leading to the hypothesis that AGN were powered by massive black holes.
The time scale varies according to the class of AGN and waveband of the spectrum. The shortest fluctuation observed in an AGN is attributed to Mrk421 with a 15 minutes duration.
However, there have been variability on time scales of months to years [21]. In the case of blazars, a rapid variability or flaring is common. Observed short time scales of a day or more in the X-ray band are characteristic to TeV blazars (e.g. Mrk421, Mrk501, PKS2155-304).
They have even substructures with shorter time scales of 10 3 − 10 5 sec [22]. On the other hand, many EGRET sources had time scales of several days [18] and the optical waveband flares are fairly slow.
After this considerations, we choose the variability time scales ranging from 15 minutes to 10 days.

IV. OUR AGN SAMPLE
Although there are more than 876 Blazars, 11777 Seyfert-1 and 48921 Quasi Stellar Objects catalogued by Verón-Cetty (2003) [23], only few of them contain the parameters important for our interests. After a thorough search of AGN catalogs, we have built our own AGN sample from some of them, so that gathering their data, we included relevant information to our investigation, such as the proper motion of jet components, luminosity and different Doppler factor estimations. From the combination of Ref. [20], the latest and widest AGN survey available on proper motion observations, with Ref. [24], which includes luminosities, and Ref. [9,11], containing Doppler factor estimations, we have selected only the objects that have information on all parameters, remaining a sample of 39 AGN. In order to get a proper motion distribution for the AGN's jets we add more data in the proper motion observations from earlier surveys [10,19,[25][26][27][28].
We must note that these multiple observations are not from the same epochs, which might introduce some uncertainties. However, we think that it is good enough for our purposes.

A. Description of the sample
Our total sample of 39 AGN, is divided into tree sub-classes: 10 BL Lacs (blazars) (see Table I), 27 Core-dominated Quasars (see Table II) and 2 Radio Galaxies (see Table III). Tables [I,II,III] show the observational data that describes the properties of each source (i.e. AGN model parameters). The data is presented as follows: • Column(1) International Astronomical Unit source designation [Name B1950].
A comment is in order, in our calculations of the neutrino flux, we need to use the UV luminosity L γ obs in Eq. (15) (see Sec. III). For that reason we extrapolate the luminosity given in this sample, which is in the radio component of the spectrum, to find the UV luminosity. This estimate has been done in the following way: from the luminosity fractions (for different parts of the electromagnetic spectrum) of the set of AGNs presented in Ref. [30], we compute the average luminosity fractions, such that the radio part represents 0.52% and the optical-UV region 26%. Then, with the relation L U V ≈ 50L rad we do the extrapolation.

B. Sample characteristics
To better understand the characteristics of the AGN sample, we present in this subsection several histograms and comparisons of its most relevant properties.
We display in Fig. 1 the angular distribution, in right ascension and declination, of our sample's 39 AGNs. Most of the sources are located in the northern hemisphere due to the abundance of observatories in that region. Nevertheless, this bias does not affect our purposes, since we are going to consider the neutrino flux as isotropic.
We show in Fig. 2  erg/sec, respectively. These observations are consistent with our expectations, since Quasars are thought to be the most luminous objects in the universe.
In Fig. 3 we present the extrapolated UV luminosity versus its intrinsic counterpart, which was obtained using Eq. (A3), for each Doppler factor estimation model. From that equation we expect that in the case of a δ slightly greater than one, the observed luminosity should be higher than the intrinsic luminosity. This situation is reflected in our sample, where the observed luminosities for each AGN is higher than their corresponding intrinsic values, which is represented by the points below the straight line. This indicates that most AGN jets in the sample are moving towards Earth and suffer the Doppler boosting. Only few AGN, in the case of the Equipartition and Inverse Compton estimation models, have the opposite behavior. This is because, for these models, some Doppler factor estimates are lower than one. On the contrary, for the Variability model all Doppler factor estimates are greater than one, meaning that their luminosities will suffer the so called Doppler boosting.
The apparent transverse velocity distribution for each knot in the sample is shown in Fig. 4. From this plot it is clear to note that Quasars have higher values of apparent transverse velocities than Blazars and Radio Galaxies, which is in agreement with Fig. 2, due to the relationship between Doppler factor, which depends on the apparent transverse velocity, and luminosity. We also notice here that some of the AGN jets are moving away from Earth, which match with our observations in Fig. 3.
The distribution of the mean angle to the line of sight, calculated using the method described in Appendix B, is given in Fig. 5 for different Doppler factor estimation models. Finally, we compare in Fig. 7, using two-dimensional plots, the different types of estimations for the Doppler factor, δ EQ , δ IC and δ V ar . We find that δ EQ and δ IC are strongly correlated, in contrast, δ V ar is uncorrelated with any of them.

V. ν µ -INDUCED EVENT NUMBER CALCULATION
We calculate the number of AGN-neutrino induced events observed on Earth within the context of the proposed IceCube [13] detector. As we are interested in astrophysical point sources (i.e. AGNs), we use muon neutrinos because they can achieve the highest angular resolution (0.7 o ) due to the large muon track, which allows to reconstruct their direction [31]. Moreover, ν µ -induced events have a higher effective volume [31] than other neutrino flavors. We restrict our search to the highest energy events in the PeV-TeV range, within IceCube's sensitivity [13], thus we avoid the background signals from atmospheric neutrinos.
Then, the number of ν µ + ν µ induced muon events is calculated by [8] N µ = T dΩ 10 18 eV where T is the exposure time (we use 10 years in our calculations) and A ef f is the effective area of the detector, taken from Ref. [15]. Since, for this energy range, IceCube is capable of observing both hemispheres, we integrate over the whole celestial sphere. Notice that in this expression, θ corresponds to the zenith angle (θ = 0 is pointing to the South Pole) and should not be confused with the angle to the line of sight used to describe the neutrino flux in the previous sections.
The total isotropic flux of AGN neutrinos summed over all sources is defined as where φ tot represents the sum of the terms F j (see Eq. (15)), after averaged over the probability density function (pdf ) in δ, defined for each j source of the total 39 AGN sources of our sample. The probability density function varies for each estimation model (mod) and is given by where γ corresponds to each Doppler factor estimation model (γ EQ = 0.49, γ IC = 0.58, γ V ar = 0.55, γ min = 0.76). The integration limits, δ maxj and δ minj , are chosen for a single AGN source, taking the maximum (δ maxj ) and minimum (δ minj ) of its corresponding set 4 of Doppler factors (these limits depend on the δ estimation model). The normalization constant C n in Eq. (21) allows us to extrapolate the total ν µ flux of our small sample to the maximum possible. This constant is given by where φ bound stands for the three different limits that we have chosen: the Waxman-Bahcall limit 3 × 10 −8 GeV −1 cm −2 s −1 sr −1 [14], the benchmark used by the IceCube Collab. 1 × 10 −7 GeV −1 cm −2 s −1 sr −1 [15] and the γ-ray bound 6 × 10 −7 GeV −1 cm −2 s −1 sr −1 [16]. We must 4 When there is only one value for an AGN source, we assume an average range.

note that the given bounds are still below the limits set by recent experiments and above
IceCube's sensitivity [15]. The limit for energies between 1 PeV and 3 EeV from the latest AMANDA results [32] derived with a 90% C.L. is E 2 Φ ν (E) < 0.99×10 −6 GeV −1 cm −2 s −1 sr −1 and the AMANDA-II-2000 data is still being processed. φ max tot at 1 GeV, is the maximum value among the set of values of φ tot obtained for all the variations performed in our work (different types of δ estimation, class of geometry emission region and variability time of the burst). It is relevant to mention that there is a possibility that these sources were optically thick enough to absorb most of the protons, only allowing neutrinos to scape, which would indeed exceed the conservative WB bound.
Continuing with the description of Eq. (20) we define P surv E νµ , θ , the survival probability of a neutrino crossing the Earth, by where χ (θ) is the column density given by with, l νµ , the neutrino distance traveled through Earth calculated using where R E is the Earth's radius (6378 Km) and d the distance to the detector (for IceCube it is 1.4 Km).
In Eq. (25) we integrate over the neutrino path crossing the Earth using the parameterized Earth's density, ρ (r), given by the Preliminary Earth Model (PREM [33]) and the inclusive interaction length, L tot int , is given by where the inclusive cross section, σ tot νN , is taken from Ref. [34], which includes the charged and neutral current contributions.
The probability that a muon produced in a ν µ charged current interaction reaches the detector before losing all its energy down to a value lower than the detector's energy threshold E thr µ (10 3 GeV for IceCube) is defined by 5 where the differential charged current cross section, dσcc dy is taken from Ref. [35] and the muon range, R µ , is taken from Ref. [36].
We include the effects of neutrino oscillations [37] by multiplying Eq. (20) by 0.5, given that the ratio of the initial to the oscillated muon neutrino flux, φ νµ : φ osc νµ , would be 2:1. It is worth mentioning that there are other scenarios where muons can lose a significant amount of energy before their decay, equivalent to considering only a single ν µ [38], otherwise we would have to consider half of the neutrino flux (without neutrino oscillations), which is different from the one we assume here.

VI. RESULTS
We present in Fig. 8, Fig. 9 and Fig. 10, an analysis of the dependence of the expected total number of induced ν µ and ν µ events versus the variability time (ranging from 15 minutes to 10 days) for different Doppler factor estimation models, emission region geometries, and for ten years of exposure in IceCube. We use the generic spectrum E −2 ν and normalize the figures with the WB limit (Fig. 8), the IceCube Collab. benchmark (Fig. 9) and the γ-ray Also, using this factor, we can comprehend two important features of our plots. First, we see why all the graphs converge to the same number of events (given for the flux limit) for an small ∆t obs , which implies that for this time we are not able to distinguish among the different model parameters under study. Second, we observe the hierarchy of the curves, where the number of events is higher for the cylinder than for the sphere geometry. This is not strange since the spherical hypothesis gives a greater volume than the cylindrical one, which implies fewer number of events given that τ ∝ 1/V . Furthermore, comparing the number of events for the same geometry, we obtain slightly higher values in the case where we average over the Doppler factor distribution; this is because the distribution includes lower values of δ, which gives, in general, a greater number of events.
In order to quantify the differences observed in our plots, at the largest variability time, in the number of events, for the different parameter hypotheses studied (e.g. type of geometries and Doppler factor calculations), we have defined the following estimator: which gives us these discrepancies in terms of the number of σ's, with N limit hyp (i) the total number of expected neutrinos, where limit stands for the bound used in the extragalactic neutrino flux and hyp (i) refers to the various hypotheses compared.
Our first analysis has the purpose to evaluate how large the differences in the number of events are for the spherical and cylindrical geometry. This is shown in Table IV, Table, we can see that, in the case of the WB limit, the differences go from 0.75σ to 1.53σ, when we do not include the δ distribution. Meanwhile, when the δ distribution is considered, these differences are even smaller going from 0.53σ to 0.89σ. For the IceCube benchmark, the differences are significant and range between 1.36-2.79σ and 1.02-1.62σ for the cases without and with δ distribution, respectively. For the γ-ray bound the differences are greater, ranging from 3.34-6.84σ and 2.35-3.98σ as described before.
In addition, we have done an analysis similar to the one before, only that in this case we have calculated the differences in the number of events considering the six different pairs of Doppler calculation's types. This is displayed in Table V and Table VI, for a specific geometry, spherical and cylindrical, respectively. There are only two favourable situations for discriminating the Doppler factor models using the number of events. This is for the case of a spherical geometry, the γ-ray bound and without δ distribution, where we have obtained 2.67σ in separation for the events predicted using δ EQ and δ IC . The other case is when we compare δ IC and δ V ar , where we have got 3.05σ. For all the rest of pair comparisons we have got negligible differences, less than 1.6σ in the splitting of the number of events.
In spite of not including variations in "a", the relation between the radius and the length in the cylinder (l = ra), we have checked that these changes will not lead to a relevant difference in the number of events.
As complementary information, we show in Fig. 11 the total number of expected events, for the three flux bounds, as a function of the slope p of the AGN spectrum, ranging from 1.3 to 2.7. It is clear that in p = 2 the number of events in this figure coincides with the maximum number of events in our precedent figures. Thus, from this plot we can rescale the results given in our tables, only multiplying the N σ by N µ (p)/N µ (p = 2), which at p = 2.7 is around √ 2.

VII. CONCLUSIONS
We have studied different neutrino production scenarios in AGNs for the highest energy neutrinos (PeV-Eev) and their impact in the total expected ν µ -induced events in the IceCube detector, due to variations in the emission region geometry, the Doppler factor estimation model and the variability time. We have obtained the total neutrino flux based on a sample of 39 AGN (i.e. point sources observations), where the limits in the total neutrino flux were the WB bound, the IceCube Collab. benchmark and the γ-ray bound.
We have found as a global behaviour that if most AGN had long times of burst, which seems natural to assume, we should be able to distinguish, in some cases, among the different models. Otherwise, for small times of burst, no information about these model parameters could be extracted through neutrino observation in a detector with the capabilities of Ice-Cube.
More specifically, for a 10-day fluctuation time and the γ-ray flux limit (6 × 10 −7 GeV −1 cm −2 s −1 sr −1 ), we could determine the emission region geometry after 10 years of IceCube exposure. We observe a separation greater than 3σ in the total number of events, between the cylindrical and spherical geometry, for all the δ cases. On the other hand, if the flux bound was below the WB limit (3 × 10 −8 GeV −1 cm −2 s −1 sr −1 ), the geometry would not introduce major changes in the total number of events. When the benchmark is 1 × 10 −7 GeV −1 cm −2 s −1 sr −1 , we are able to discriminate between geometries in half of the δ cases, without including the δ distribution, with a separation in the total number of events greater than 2σ.
It is important to note that it is possible to infere from Fig. 8, Fig. 9 and Fig. 10, any intermediate case (a mixture of both geometries), between the cylindrical and spherical geometry for the emission region. In this sense, how far or close we are from some of these extreme behaviours (purely cylindrical or spherical) is a measure of the tendency of the sample.
For the Doppler factor estimation models, in most cases, the total number of neutrino induced events gives no relevant information or, in other words, the differences in δ do not affect this number. Nonetheless, for a spherical emission region, ∆t obs =10 days, the γray bound and without considering the δ distribution, it is possible to distinguish between δ EQ -δ IC and δ IC -δ V ar obtaining an event separation greater than 2.5σ.
It is relevant to keep on mind that our analysis has considered that the neutrino spectrum had slope p = 2. Therefore, for a slope p > 2, the separation in terms of N σ must increase, which means that we are going to enlarge the number of cases where we can differentiate the geometries and the Doppler factor estimations models.
As closing remarks, we must say that our work establish possible tendencies in the parameters in the AGNs. In fact, we expect that these tendencies become more robust and measurable in the future, when we have a bigger sample containing the necessary information and accurate point sources observations.

APPENDIX A: RELATIVISTIC TRANSFORMATIONS
For the energy and time transformation between rest-frame to observer-frame, which includes not only the common relativistic transformation but also the cosmological expansion, we use where δ is the relativistic Doppler factor, also known as bulk Lorentz factor and z the AGN There are luminosity transformations [17] which consider the observed integrated spectrum and solid angle distributions for moving, isotropic sources and for a continuous standard conical jets. However, in order to proceed in a more general way (i.e. model independent) we consider the following jet photon luminosity transformation First, we calculate the apparent transverse velocity β app (j, k) corresponding to each knot (index k) for its AGN source (index j), given by 7 6 There should not be any misleading impression with the fact that the redshift means a receding object and most Doppler factors values an approaching source, because the redshift is due to cosmological effects and the Doppler factor to the moving knots in the jets. 7 The formula from Pearson&Zensus (1987) widely used in older papers is replaced by Eqs. (B1,B2) which consider a non-zero cosmological constant.
where µ(j, k) is the proper motion for each knot in the source, d m (j) is the "proper motion distance" to the source, calculated using [39] with the Hubble constant H 0 = 71 Km s −1 Mpc −1 , the matter density Ω m = 0.27 ± 0.04, and dark energy density Ω Λ = 0.73 ± 0.04, according to WMAP latest results [40].
Then, we calculate the angle to the line of sight, θ(j, k) mod , for each proper motion component and its corresponding Doppler factor for each estimation model (the index mod = EQ, IC, Var), using In the case of Minimum Doppler factor, which gives the minimum angle, we use ¿From the distribution of θ(j, k) mod for each source, we obtain its mean value θ (j)mod and assign it to the source, which is well justified [24]. Thus, we can construct the Doppler factor distribution for each model in terms of the Doppler factor of the knots for each AGN source. where We discard Doppler factors that could correspond to jets affected by significant bends, twists and inward moving features.        Note.-The acronyms used in the references for the proper motion, given in column (10)