Event rates and flux limits for candidate source

In this section, flux limits are computed for candidate sources where theoretical models have been published. Table 5 lists the selected candidate sources and the number of expected events with the specified model. Equation 9 was used for the calculation of $N_{\mu}$. Szabo and Protheroe assume that all the protons leaving the core of the AGN interact with surrounding photons, while Stecker et al. assumes that protons will not interact if the straight-line optical depth to proton-photon collisions is less than one [9]. These two models of hadron acceleration represent extremes in the expected neutrino fluxes from NGC4151. For Mrk501 results, the time averaged differential photon flux measured by HEGRA [10] was used to estimate the differential neutrino flux ( $\frac{dN}{dE} = N_{0} (E/1$ TeV $)^{-\gamma} \cdot e^{-\frac{E}{E_{0}}}$ with $N_{0}=10.8\cdot10^{-11}$ cm^-2s^-1TeV^-1, $\gamma=1.92$, and E₀=6.2<> TeV). In the second model for Mrk501 shown in table 5, the exponential cut-off term was eliminated. This may apply if the higher energy photons are absorbed by the infrared photons. Then, the energy spectrum of the neutrino traces the unattenuated differential energy spectrum of the source.

 

Table 5: Number of expected events from three different sources with a live-time of 138.2 days. The different models are discussed in the text.

Source	Model	$N_{\mu}$
Mrk501	$\Phi_{\nu}=\Phi_{\gamma}$ (HEGRA) $0.01<E_{\nu}<100$ TeV	0.145
Mrk501	$\Phi_{\nu} \propto E_{\nu}^{-1.92}$ (HEGRA) $1<E_{\nu}<10^{3}$ TeV	1.12
Mrk421	Szabo and Protheroe $1<E_{\nu}<10^{3}$ TeV	0.0404
NGC4151	Szabo and Protheroe $1<E_{\nu}<10^{3}$ TeV	0.237
NGC4151	Stecker $1<E_{\nu}<10^{3}$ TeV	0.0171

Table 5 shows that these particular models are not testable by AMANDA B10 for the 138.2 days of live-time. Yet the assumption of pure power law with a spectral index of 1.92 for Mk501 yields a predicted signal that may be testable in the near future. The sky bin surrounding Mk501 contains four data events. Hence a reduction of background by a factor of 4 to 5, may test this particular model for Mrk501. This is an encouraging result, showing the analysis is not far from making constraints on certain acceleration models of astrophysical sources. The neutrino and muon flux limit for these sources were determined by using an optimal circular bin with the candidate point source at the center. The number of data events within the circular bin was used to determine the fluctuation expected from background consistent with 90% CL, $\mu (N_{0} , N_{b})$. Equation 11 was used to calculate the flux limit. Table 6 shows both the neutrino and muon flux limits for a number of candidate point sources.

 

Table 6: Muon and neutrino flux limits on candidate point sources. All limits are with a 90% CL.

Source	Model	bin size	$\Phi_{\mu}^{limit}$ (cm-2 sec-1)	$\Phi_{\nu}^{limit}$ (cm-2 sec-1)
Mrk501	$\Phi_{\nu}=\Phi_{\gamma}$ (HEGRA) $0.01<E_{\nu}<100$ TeV	5.4o	2.25x10-14	0.0363x10-9
Mrk501	$\Phi_{\nu} \propto E_{\nu}^{-1.92}$ (HEGRA) $1<E_{\nu}<10^{3}$ TeV	5.4o	6.12x10-14	0.482x10-9
Mrk421	Szabo and Protheroe $1<E_{\nu}<10^{3}$ TeV	5.8o	1.04x10-14	1.45x10-9
NGC4151	Szabo and Protheroe $1<E_{\nu}<10^{3}$ TeV	5.0o	1.08x10-14	1.50x10-9
NGC4151	Stecker $1<E_{\nu}<10^{3}$ TeV	5.0o	0.61x10-14	0.086x10-9
Cygnus X-3	E-2 spectra $1<E_{\nu}<10^{3}$ TeV	5.4o	1.36x10-14	1.84x10-9
Hercules X-1	E-2 spectra $1<E_{\nu}<10^{3}$ TeV	5.5o	1.14x10-14	1.44x10-9
Crab Nebula	E-2 spectra $1<E_{\nu}<10^{3}$ TeV	4.6o	1.61x10-14	1.89x10-9
Geminga	E-2 power spectra $1<E_{\nu}<10^{3}$ TeV	5.8o	1.80x10-14	2.11x10-9

The number of experimental data events in these bins was between 4 to 12 giving a 90% CL in the background fluctuation between 4.6 and 7. The efficiency was between 0.662 and 0.765 depending on the source and the angular bin size. Note the flux limits calculated are highly model dependent, as was shown generically in figure 27 and figure 28. Figure 29 compares this analysis results with the best published limits of muon flux. The MACRO neutrino detector is located in Gran Sasso tunnel, Italy. The muon energy threshold is about 1 GeV, while it is greater than 10 GeV for AMANDA. Crab Nebula and Geminga are two sources low in the sky with a declination of 22 and 18 degrees respectively. Due to the more favorable declination of 38 degrees, the flux limit for Mk421 is lower than the best published limit [11] by a factor of 5. The muon flux limit for the Crab Nebula are approximately as MACRO due to its location near the horizon. Since the position of the galactic plane are close to the horizon of the AMANDA detector many interesting candidate sources are near the horizon. This provides strong motivation for optimizing the analysis to produce greater sensitivity near the horizon. The current point source analysis shows good progress for achieving this goal.

The AMANDA-B10 detector has maximum sensitivity for the near vertical positive declinations, which complements the sky regions scanned by MACRO which achieves maximum sensitivity for declination less than 20 degrees, and Super-Kamiokande which achieve maximum sensitivity for source with declinations less than +50 degrees (see Figure 30) . The existing limits are rather good. Super-Kamiakande has been operating for about 4 years, and will soon report negative search limits for 4000 m²<>yr exposure factors. MACRO has about half that (they are smaller, but have operated longer). The AMANDA point source analysis provides more uniform sky coverage than previously reported (Kim, ICRC) (figure 26), and the sensitivity is greater. AMANDA closed the last hole in the sky, and we rule out the hope that it may have contained a source that is substantially brighter than the flux limits that currently exist at negative declinations. The analysis provides an initial attempt to correct for inefficiencies due to finite angular resolution, trials factors and other statistical penalties introduced by this particular analysis, etc., but significant work remains.

    

Figure 29: of muon flux limits given by MACRO [11] and from this work. Horizontal arrows indicate the approximate energy threshold and the vertical arrows the flux limit.
Figure 30: flux limits versus declination angle for our analysis (AMANDA-B10) with one year of data, MACRO [11] with 10 years of data, and anticipated value for Super-Kamiokande [12] with 4 years of data.