Appendix B: More on Mkn 421 spectra

 

 

Table B.1: Mkn 421 available statistics.

1998

2000 (except 4-5/02)

4-5/02/2000

${T_{{\rm ON}}\,=\,6.2\:{\rm h}}$

${\theta_{\rm z}\,\in\,[0^\circ;28^\circ]}$

${T_{{\rm ON}}\,=\,8.4\:{\rm h}}$

${\theta_{\rm z}\,\in\,[0^\circ;28^\circ]}$

${T_{{\rm ON}}\,=\,3.5\:{\rm h}}$

${\theta_{\rm z}\,\in\,[0^\circ;28^\circ]}$

${T_{{\rm OFF}}\,=\,9.5\:{\rm h}}$

${S_\gamma\,=\,735\,\pm\,57}$

${T_{{\rm OFF}}\,=\,16.7\:{\rm h}}$

${S_\gamma\,=\,1424\,\pm\,71}$

${T_{{\rm OFF}}\,=\,17.6\:{\rm h}}$

${S_\gamma\,=\,609\,\pm\,40}$

${\Delta_{i_{\rm e}}}$

${n_{i_{\rm e}}}$

${p_{i_{\rm e}}}$

${S_{i_{\rm e}}}$

${\delta S_{i_{\rm e}}}$

${{N_\sigma}_{i_{\rm e}}}$

${n_{i_{\rm e}}}$

${p_{i_{\rm e}}}$

${S_{i_{\rm e}}}$

${\delta S_{i_{\rm e}}}$

${{N_\sigma}_{i_{\rm e}}}$

${n_{i_{\rm e}}}$

${p_{i_{\rm e}}}$

${S_{i_{\rm e}}}$

${\delta S_{i_{\rm e}}}$

${{N_\sigma}_{i_{\rm e}}}$

${0.3}-\scriptstyle{0.5}$

${615}$

${445}$

${170}$

${29}$

${5.8}$

${1351}$

${993}$

${358}$

${44}$

${ 8.2}$

${464}$

${318}$

${146}$

${23}$

${ 6.4}$

${0.5}-\scriptstyle{0.8}$

${790}$

${536}$

${254}$

${35}$

${7.3}$

${1267}$

${731}$

${536}$

${41}$

${13.1}$

${561}$

${300}$

${261}$

${25}$

${10.4}$

${0.8}-\scriptstyle{1.3}$

${452}$

${301}$

${151}$

${26}$

${5.7}$

${ 690}$

${411}$

${279}$

${30}$

${ 9.2}$

${300}$

${172}$

${128}$

${18}$

${ 7.0}$

${1.3}-\scriptstyle{2.0}$

${229}$

${123}$

${107}$

${18}$

${5.9}$

${ 335}$

${158}$

${175}$

${21}$

${ 8.6}$

${138}$

${ 64}$

${ 74}$

${12}$

${ 6.0}$

${2.0}-\scriptstyle{5.0}$

${119}$

${ 66}$

${ 53}$

${13}$

${4.0}$

${ 165}$

${ 89}$

${ 76}$

${15}$

${ 5.1}$

${-}$

${-}$

${-}$

${-}$

${-}$

 

 

Table B.2: Covariance matrices elements of Mkn 421 spectra.

Period	${V_{\phi\phi}^{\rm pl}}$	${V_{\phi\gamma}^{\rm pl}}$	${V_{\gamma\gamma}^{\rm pl}}$	${V_{\phi\phi}^{\rm cs}}$	${V_{\phi\gamma}^{\rm cs}}$	${V_{\phi\beta}^{\rm cs}}$	${V_{\gamma\gamma}^{\rm cs}}$	${V_{\gamma\beta}^{\rm cs}}$	${V_{\beta\beta}^{\rm cs}}$
${1998}$	${3.97\,\times\,10^{-2}}$	${-1.24\,\times\,10^{-2}}$	${1.44\,\times\,10^{-2}}$	${1.03\,\times\, 10^{-1}}$	${-1.10\,\times\, 10^{-2}}$	${1.13\,\times\, 10^{-1}}$	${2.35\,\times\, 10^{-2}}$	${3.28\,\times\, 10^{-2}}$	${2.35\,\times\, 10^{-1}}$
${2000}$	${3.19\,\times\, 10^{-2}}$	${-8.65\,\times\, 10^{-3}}$	${6.51\,\times\, 10^{-3}}$	${7.80\,\times\, 10^{-2}}$	${-8.36\,\times\, 10^{-4}}$	${6.45\,\times\, 10^{-2}}$	${1.68\,\times\, 10^{-2}}$	${2.91\,\times\, 10^{-2}}$	${1.36\,\times\, 10^{-1}}$
${4-5/02/2000}$	${1.12\,\times\, 10^{-1}}$	${-3.84\,\times\, 10^{-2}}$	${2.16\,\times\, 10^{-2}}$	${1.38\,\times\, 10^{-1}}$	${-3.45\,\times\, 10^{-2}}$	${3.63\,\times\, 10^{-2}}$	${1.53\,\times\, 10^{-1}}$	${2.86\,\times\, 10^{-1}}$	${6.57\,\times\, 10^{-1}}$

   
Appendix B.1: Available statistics

The statistics used for extracting the Mkn 421 spectra are given in Table B.1. With the same notations used in this paper, we write:

• the data zenith angle amplitude, and the ON and OFF-source observation durations, $T_{{\rm ON}}$ and $T_{{\rm OFF}}$;
• for each of the $n_{\rm e}$ energy intervals $\Delta_{i_{\rm e}}\equiv[\widetilde{E}^{{\rm min}}_{i_{\rm e}}, \widetilde{E}^{{\rm max}}_{i_{\rm e}}]$ (in ${\rm TeV}$):

  $\bullet$ the numbers of events passing the selection cuts in the ON and OFF data,
  $n_{i_{\rm e}}=\sum_{i_{\rm z}}n_{i_{\rm z},i_{\rm e}}$ and $p_{i_{\rm e}}=\sum_{i_{\rm z}}\beta_{i_{\rm z}}p_{i_{\rm z},i_{\rm e}}$;

  $\bullet$ the observed number of $\gamma$ events, $S_{i_{\rm e}}=n_{i_{\rm e}}-p_{i_{\rm e}}$, and its error, $\delta S_{i_{\rm e}}={\left({\sum_{i_{\rm z}}n_{i_{\rm z},i_{\rm e}}+ \beta_{i_{\rm z}}^2p_{i_{\rm z},i_{\rm e}}}\right)}^{1/2}$;

  $\bullet$ the statistical significance ${N_\sigma}_{i_{\rm e}}=S_{i_{\rm e}}/ \delta S_{i_{\rm e}}$;

• the observed total number of $\gamma$ events, $S_\gamma=\sum_{i_{\rm e}}S_{i_{\rm e}}$, and its error, $\delta S_\gamma={\left({\sum_{i_{\rm e}}\delta S_{i_{\rm e}}^2}\right)}^{1/2}$.

   
Appendix B.2: Covariance matrices

Let $V^{\rm pl}$ and $V^{\rm cs}$ be the two covariance matrices of spectral parameters for the $\mathcal{H}^{\rm pl}$ and $\mathcal{H}^{\rm cs}$ hypotheses (see Sect. 2.3), respectively:

$$V^{\rm pl}=\left(\begin{array}{cc} V_{\phi\phi}^{\rm pl}&V_{\... ...\beta}^{\rm cs}&V_{\beta\beta}^{\rm cs}\\ \end{array}\right).$$

The values of these matrices elements are given in Table B.2 for each spectrum of Mkn 421 obtained in this paper.

In the power-law hypothesis, the energy $E_{\rm d}$ at which the values of $\phi_0^{\rm pl}$ et $\gamma^{\rm pl}$ are decorrelated writes:

$$E_{\rm d}=\exp\left[\frac{V_{\phi\gamma}^{\rm pl}}{\phi_0^{\rm pl}V_{\gamma\gamma}^{\rm pl}}\right]\:{\rm TeV}.$$

At this decorrelation energy, the width of the hatched area representing the 68% confidence level contour, is minimum: for instance, we find $E_{\rm d}=690~{\rm GeV}$ for the 1998 time-averaged spectrum of Mkn 421 shown in Sect. 3.2.

In the curved shape hypothesis, the energy-dependent exponent $\gamma^{\rm cs}_l(E_{{\rm TeV}})=\gamma^{\rm cs}+\beta^{\rm cs}\log_{10}E_{{\rm TeV}}$ has a minimal error at the energy $$E^{\rm cs}_0=10^{\frac{-V_{\gamma\beta}^{\rm cs}}{V_{\beta\beta}^{\rm cs}}}\:{\rm TeV}$$; the corresponding value of $\gamma^{\rm cs}_l$ writes

$$\gamma^{\rm cs}_0\equiv\gamma^{\rm cs}_l(E^{\rm cs}_0)=\gamma... ...m cs}\frac{V_{\gamma\beta}^{\rm cs}}{V_{\beta\beta}^{\rm cs}},$$

and its error

$$\delta\gamma^{\rm cs}_0\equiv\delta\gamma^{\rm cs}_l(E^{\rm c... ...c{{V_{\gamma\beta}^{\rm cs}}^2}{V_{\beta\beta}^{\rm cs}}}\cdot$$