On Integral Upper Limits Assuming Power Law Spectra and the Sensitivity in High-Energy Astronomy

0.61 MB·

by  Max L. Ahnen

Draft version January 24, 2017
PreprinttypesetusingLATEXstyleAASTeX6v.1.0
ON INTEGRAL UPPER LIMITS ASSUMING POWER LAW SPECTRA
AND THE SENSITIVITY IN HIGH-ENERGY ASTRONOMY
Max L. Ahnen1
InstituteforParticlePhysics,ETHZurich,8093Zurich,Switzerland
[email protected]
7
1 ABSTRACT
0
The high-energy non-thermal universe is dominated by power law-like spectra. Therefore results in
2
high-energy astronomy are often reported as parameters of power law fits, or, in the case of a non-
n
detection, as an upper limit assuming the underlying unseen spectrum behaves as a power law. In
a
J this paper I demonstrate a simple and powerful one-to-one relation of the integral upper limit in the

1 two dimensional power law parameter space into the spectrum parameter space and use this method
2 to unravel the so far convoluted question of the sensitivity of astroparticle telescopes.

Keywords: gamma-rays: general — methods: statistical
]
M
I 1. INTRODUCTION tronomy. However, the universal observation of power
.
h lawsingamma-raysourcesallowsframingthequestions
Power law spectra are a general feature in the non-
p
in the {f ,Γ} parameter space of the power law.
- thermal high-energy universe. Often results in high- 0
o
energy astronomy are reported as parameters of power
r
t law fits (e.g. Ackermann et al. 2016; Aharonian et al.
s
a 2006b), or, if no source is detected, as a flux limit (e.g.
[ Archambault et al. 2016; Aliu et al. 2012, 2014a; Ah-

1 nen et al. 2016b; Aleksi´c et al. 2015; Abramowski et al. 2. UPPER LIMITS IN THE CONTEXT OF ON/OFF
v 2014;Aliuetal.2015;Ahnenetal.2016a)assumingthe MEASUREMENTS
8
unseen spectrum dN/dE is a power law
4 Inthelowcountregimeofhigh-energyastronomyone
0 dN (cid:18) E (cid:19)Γ experimental method is well established: The On/Off
06 dE =f0 E . (1) measurement(Li&Ma1983;Rolkeetal.2005;Knoetig
0
2014; Abraham et al. 2007; Abbasi et al. 2011). In it,
.
1
In Eqn. 1 E is a freely chosen reference energy, f is the experimenter would like to measure an imprecisely
0 0 0
the flux normalization, and Γ is the spectral index. known background count rate as well as a potential sig-
7
1 The ability of gamma-ray, neutrino, and cosmic-ray nal count rate. The measurement consists of two ob-
: telescopes to detect a source depends on their intrin- servations: The first counts N events in a region with
v on
i sic parameters and on the external source spectrum potential signal, the second counts Noff events in an ad-
X
(Aleksi´c et al. 2016; Aharonian et al. 2006d; Murase & jacent region which is assumed to be signal free. Both
r Takami 2008; Bertou et al. 2002; Aartsen et al. 2015). regions are related to one another through the ratio of
a
However, even when no significant signal counts are de- exposures α. If the data show no evidence for a source,
tected, lone would like to put limits on the observed an upper limit on the signal counts λ can be calcu-
lim
signal flux would allow inference of physical results, as latedusingN ,N ,andα (Rolkeetal.2005;Knoetig
on off
observing nothing is fundamentally different from not 2014). Thislimitcanbeusedtoinferconstraintsonthe
observing at all. But what does an upper limit on the potential source spectrum.
signal count rate mean for the spectrum? This ques- The most constraining upper limits can be extracted
tion is tightly related to the detection limit of a given from integral upper limits, calculated using the average
instrument. After all, every observer is interested to signal counts λ integrated over all energies. Then, in
s
know: How long does it take to detect my source with order to to make inferences about the source spectrum
an assumed average flux? one, usually, further assumes a certain power law with
Currently,thereareonlyconvolutedorpartialanswers a common spectral index Γ (Abramowski et al. 2014;
to these two fundamental questions in high-energy as- Aleksi´c et al. 2015; Ahnen et al. 2016b; Aliu et al. 2015,
2 Ahnen
2014a). variations in Γ, it is still not universal as three common
(cid:90) dN power law indices have to be chosen, none of which are
λs=t dE Aeff(E)dE, (2) distinguished.
In this paper, I demonstrate a more useful approach
(cid:90) (cid:18) E (cid:19)Γ
=tf A (E)dE. (3) to integral upper limits than the previously discussed
0 E0 eff methods. The key insight comes from investigating in-
Assuming a steady source, the integrated average sig- tegral upper limits in the two dimensional {f0,Γ} pa-
nal counts λ depend on the instrument effective area rameter space of the power law.
s
for each energy A (E), the observation time t, and the
eff
assumed source spectrum parameter Γ. The limit flux 3. GENERALIZED INTEGRAL UPPER LIMITS
ASSUMING POWER LAW SPECTRA
normalizationf isthencalculatedsuchthattheresult-
0
ing power law spectrum would yield the limit InthefollowingIderivethenovelintegralupperlimit
method, and compare it to, a data set from the VER-
λ =λ . (4)
s lim
ITAS Cherenkov telescope (McCutcheon 2012). In it,
The result is often reported as integrated source flux upper limits from an observation of the globular cluster
F above a certain threshold energy E M13 are reported. M13 is measured to host millisecond
>Ethr thr
(cid:90) ∞ dN pulsars (Hessels et al. 2007) and is a popular, yet un-
F>Ethr = dE dE. (5) detected,targetforCherenkovtelescopes (McCutcheon
Ethr 2012; Anderhub et al. 2009). In this case the observa-
Implicit in this method is the choice of the power law tions took place during the VERITAS epoch V5 (Park
indexΓwhichcanbeextrapolated(Ahnenetal.2016b), 2016) in 2010. The following components must be spec-
physicallymotivated (Aliuetal.2015,2014a),orhedged ified to calculate any integral upper limit for a specific
to have little impact on the result (Abramowski et al. On/Off measurement:
2014;Aleksi´cetal.2015). However,extrapolatingisnot
always possible and the power law index often varies 1. Aeff(E): Effectiveareaasafunctionoftrueenergy
strongly from source to source (Ackermann et al. 2016), after all cuts
whichpropagatestoapotentiallylargesystematicerror
2. t: Effective observation time
in the flux limits.
Amethodtoreducethesystematicerrorofthechoice
3. λ : Choiceofcriteriononthesignalcountslimit
lim
of power law index and to infer spectral information in
e.g., Rolke et al. (2005); Knoetig (2014); calcula-
the absence of signal is to calculate differential upper
tion relies on
limits. Although they are called differential, no obser-
vation can measure differentials directly. Therefore, dif- (a) Choice of credible interval or confidence in-
ferential upper limits are approximated by upper limits terval e.g., 95%,99%
integrated in small bins, assuming a certain power law
(b) α: On/Off region exposure ratio
index Γ and using Eqn. 3 and Eqn. 4. The limit curve
obtainedinthiswaystilldependsonthechoiceofΓ(Ah- (c) Non: Number of events in the On region
nen et al. 2016b), admittedly less than the full integral
(d) N : Number of events in the Off region
off
upperlimit. Themainissuewithdifferentialupperlim-
its is their lower sensitivity due to the binning, which ThecomponentsusedinthissectionareshowninTab.1.
falls short of the real detection capabilities of the in- The resulting signal limit λ is calculated using the
lim
strument. 95% credible limit on the signal posterior using the
A more sensitive approach to the influence of method from Knoetig (2014).
the power law index Γ is the decorrelation energy When investigating spectra of sources it is important
method (Aliu et al. 2012; Archambault et al. 2016). to take the difference between estimated energy E
est
Here, integral upper limits are first calculated assuming and true (or simulated true) energy E into account.
three different power law indices. This results in three This relationship is the migration matrix. This means
different limit power laws. The decorrelation energy is that the effective area measured over estimated energy
thendefinedastheenergyatwhichtheirfluxandthere- A (E )isaconvolutionoftheeffectiveareaovertrue
eff est
fore their upper limit estimate depends the least on the energy A (E), the migration matrix, and the source
eff
powerlawindex. Inafinalsteptheweakestofthethree spectrum. However, as the method proposed here is an
upper limits at the decorrelation energy (which turns integral limit method, it needs the effective area as a
out to be the upper limit of the central power law in- function of the true (simulated) energy A (E). In my
eff
dex) is reported. While this method better accounts for case however, I use the commonly reported A (E )
eff est
Integral upper limits assuming power law spectra 3
VERITAS V5 λs(f0,Γ), t = 7h, E0 = 1000 GeV
a) 10-9
b)
2.0 11.3 10-10
0 10-11
3.
2.5 1]−10-12
V)
e
Γ 5.0 2m s T10-13
3.0 E / [(c10-14
d
N/10-15
3.5 8.0 17.0 25.0 38.0 d10-16
10-17
10-18
0.5 1.0 1.5 2.0 2.5 3.0 3.5 102 103
f0 / [(cm2 s TeV)−1] 1e 13 E / GeV
Figure 1. a) The average signal counts as a function of the assumed power law parameters λs(f0,Γ) (see Eqn. 3) for a real
observation example from the VERITAS Cherenkov telescope (McCutcheon 2012). The solid line marks λ . The power laws
lim
on this line represent the border of the excluded parameter space (to the right of the solid line). Dots indicate example power
laws. b)ExamplepowerlawsdrawnfromthefamilyofcurvesontheVERITASM13exclusionlinerepresentedinthespectrum.
Itbecomesapparentthatthefamilyofpowerlawsontheexclusionlineproducesanenvelopecurvewhenregardingfluxupper
limits in the spectrum parameter space.
(McCutcheon 2012; Aleksi´c et al. 2016) as an approxi- lawparameterspace(Ackermannetal.2016). However,
mationtoA (E)asnosuitableeffectiveareasarepub- most physicists compare source spectra and models in
eff
lishedtogetherwiththenecessarydatacomponents. the spectrum parameter space.
The following translation of the implicitly defined
Table 1. VERITAS M13 observations and λ cal- power law exclusion curve (Fig. 1a) into the spectrum
lim
culation parameter space (Fig. 1b) is key. When looking at a set
ofpowerlawsfromthefamilyofcurvesontheexclusion
line,itbecomesapparentthattheyproduceanenvelope
A t C.I. α N N λ
eff on off lim
curve when regarding flux upper limits in the spectrum
(h)
parameter space. This means that for every assumed
1) 7 0.95 1/10 55 670 11.3 power law index Γ there is an energy I call sensitive en-
ergy E (Γ) in the spectrum parameter space. At the
sens
sensitiveenergyE (Γ)thelimitpowerlawwithΓhas,
Note—The On/Off measurement parameters from sens
locally,themaximumfluxdN/dE(Eqn.1),comparedto
Tab.5.2andFig.5.10ofMcCutcheon(2012). The
λ critereonchoseninthispaperisfromKnoetig the other power laws on the power law exclusion curve.
lim
(2014) using a 95% credible interval. The calcu- Therefore, one specific integral flux upper limit is dis-
lated λ is shown in bold.
lim tinguished at E (Γ), which can be used to construct
sens
References—1) McCutcheon (2012) a one-to-one relation from points on the power law ex-
clusioncurvetopointsinthespectrumparameterspace
by
(cid:18) (cid:19)
dN
(E)=
When examining the family of excluded power laws dE
lim
in the {f ,Γ} parameter space for the VERITAS M13
0 dN (6)
observation, the result is Fig 1a). It shows the aver- maximize (E),
age signal counts as a function of the power law pa- f0,Γ dE
subject to λ =λ .
rametersλ (f ,Γ)(Eqn.3),togetherwiththeimplicitly s lim
s 0
defined curve where the average signal counts λ are The curve (dN/dE) (E) in the spectrum parameter
s lim
equal to the signal limit λ (Eqn. 4). The power laws space represents the border of the region, which I call
lim
on this curve represent the border of the excluded pa- integral spectral exclusion zone. The necessary condi-
rameter space — the region with higher fluxes to the tions for the existence of the integral spectral exclusion
rightisexcluded. Thisplotcanalreadybeusedtocom- zonecanbededucedfromthemethodofLagrangemul-
pare models with limits, as sources populate the power tipliers, which is done in Appendix A. A fundamental
4 Ahnen
104 sensitive Energy Esens(Γ) 10-8 VERITAS V5 M13 UL comparison t = 7h
VERITAS V5 a) MAGIC C.U.
ApJ 674
10-9
0.1 C.U
0.01 C.U
10-10
diff. U.L.
1]− int. U.L.
V)10-11
e
V T
E / Gesens103 2dE / [(cm s 1100--1132
N/
d
10-14
10-15
b)
102 10-16
5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 102 103 104
Γ E / GeV
Figure 2. a)Thesensitiveenergy Esens(Γ)giventheeffectiveareaofFig5.10inMcCutcheon(2012). Harderspectraarebetter
constrained at higher energies, softer spectra are better constrained at lower energies. b) The integral spectral exclusion zone
(solid line), marking the physical boundary above which every power law would have been detected, independent of its index
Γ. It is shown in comparison to the reported limits from McCutcheon (2012). The reported differential upper limits are about
five times weaker than the integral spectral exclusion zone. The VERITAS integral flux limit at the decorrelation energy is
compatible with the integral spectral exclusion zone at this one point. Furthermore, the spectrum of the Crab nebula (Albert
et al. 2008) is shown for comparison.
analytical result of this is 2. Calculate Eqn. 6 by maximizing the local flux un-
der constraints directly.
E (Γ)=exp(µ(Γ)), (7)
sens
whereµ(Γ)isdefinedastheaverageofthenaturalloga-
What is the physical meaning of E (Γ) and the in-
sens
rithm of the true energy E over sensitive area weighted
tegralspectralexclusionzone? Incaseapowerlawwith
by the power law
index Γ crosses into the integral spectral exclusion
cross
(cid:82) EΓA (E)ln(E)dE zone,itscorrespondingimplicitlydefined(Eqn.4)upper
eff
µ(Γ):= (cid:82) . (8)
EΓA (E)dE limitfluxnormalizationf (Γ )atthesensitiveenergy
eff 0 cross
E (Γ ) is lower. Therefore, the border of the in-
The example VERITAS E (Γ) is shown in Fig. sens cross
sens
tegral spectral exclusion zone is the physical boundary
2a). To calculate the integral spectral exclusion zone
where every source with a power law spectrum cross-
(dN/dE) (E), one has to determine the power law
lim ing it would be detected, independent of Γ . From
{f(cid:48),Γ(cid:48)}thatislocallytangenttotheborderofthezone cross
0 this point of view, E (Γ ) is the energy at which
at energy E. This can be done in two ways: sens cross
a source with a given power law index Γ is best
cross
1. Use the Lagrange multiplier results. constrained. As the sensitive energy is a monotonically
increasing function of Γ (see Appendix A), this yields
(a) Calculate Γ(cid:48) numerically by inverting E =
the intuitive result that harder spectra are better con-
E (Γ(cid:48)) (using Eqn. 7).
sens
strained at higher energies and that softer spectra are
(b) Calculate f(cid:48) by solving the power law exclu-
0 better constrained at lower energies.
sion line constraint Eqn. 4
Applying this method to the VERITAS M13 data set
f(cid:48) = λlim , (9) results in Fig 2b). Here, the integral spectral exclu-
0 tc(Γ(cid:48)) sion zone is calculated maximizing the logarithm of the
where c(Γ(cid:48)) is defined as the spectrum- flux under constraints directly (see Eqn. 6) using the
weighted acceptance scipy.optimize Python library (Jones et al. 2001–). The
integral spectral exclusion zone is compatible with the
c(Γ(cid:48)):=(cid:90) (cid:18) E (cid:19)Γ(cid:48)A(E)dE. (10) reportedintegrallimitatthedecorrelationenergyandis
E0 about5timesmoresensitivethanthereporteddifferen-
tial limit. The VERITAS method of reporting an inte-
(c) Insert parameters in
gral upper limit at the decorrelation energy (see Sec. 2)
(cid:18) (cid:19)
dN dN
(E)= (E)| . (11) approximatestheintegralspectralexclusionzoneatone
dE lim dE f0(cid:48),Γ(cid:48) point, which is also apparent from the method. In this
Integral upper limits assuming power law spectra 5
sense the integral spectral exclusion zone is a general- for selected and specific analyses. This means that it
ization of the decorrelation energy method. is not possible, outside of a telescope collaboration, to
calculate the time until an analysis detects a power law
4. APPLICATION: THE SENSITIVITY OF source without further assumptions. I therefore make
ASTROPARTICLE TELESCOPES the following assumptions (see Tab 2) in order to use
Oneimportantapplicationoftheintegralspectralex- the published MAGIC results (Aleksi´c et al. 2016):
clusion zone is the fundamental question: What sources
1. α = 1/5: Possible values for α stated in that
can a certain instrument detect? After all, upper lim-
publication are 1/3 and 1/5. Given that similar
its are a statement of sensitivity. Therefore, the current
Cherenkov telescopes use even more Off regions
methodstacklingthisquestionareequivalenttothecur-
(see Tab. 1), I assume the more sensitive value
rent methods of calculating upper limits described in
1/5 is reasonable.
Sec. 2.
First, there are integral methods assuming a certain
2. A (E): The reported effective area is not given
powerlawindexΓ. Amongthepossiblepowerlaws, the eff
after all cuts. The authors used an additional
power law with minimum flux normalization f which
0
energy cut at ∼ 300GeV to achieve the most
is required for a detection in a certain observation time
sensitive instrument. Therefore, to approximate
is chosen. This is often reported as a function of ob-
A (E), I use the reported A (E ) with a
servation time (Aharonian et al. 2006d). A similar ap- eff eff est
threshold at 300GeV convoluted with the energy
proach is to report the integral sensitivity as a function
resolution stated in Aleksi´c et al. (2016).
oftheminimumfluxrequiredfordetectionaboveagiven
threshold energy (Aleksi´c et al. 2016). However, this
Table 2 further contains the MAGIC analyses specific
method does not constitute a full analysis as the last zenith distance domain names (low Zd ∈ [0◦,30◦] and
cut (in energy) is unspecified. Therefore, it can not be medium Zd∈[30◦,45◦]) and background rates σ .
bg
compared to other sensitivities easily. Furthermore, the
power law index problem persists: What if the source
Table 2. Inputs for sensitivity calculations of the
has a different Γ?
MAGIC Cherenkov telescope
Second, authors report sensitivities as differential up-
per limits (Aleksi´c et al. 2016; Bernl¨ohr et al. 2013),
solving the power law index issue. This method is also mode A α σ extra cut
eff bg
used to compare instrument sensitivities (Funk et al.
(1/h)
2013) but, as discussed in Sec 2, being a differential
low Zd 1) 1/5 7.37 E >300GeV
limit, it falls short of the real detection capabilities of est
med. Zd 1) 1/5 28.83 E >300GeV
the instruments. est
I demonstrate my novel method using the latest pub-
lished performance data from the MAGIC Cherenkov
Note—MAGIC published low zenith distance param-
telescope(Aleksi´cetal.2016). ForagivenOn/Offmea- eters (Zd ∈ [0◦,30◦]) and medium zenith distance
surement analysis, the following components define the parameters (Zd ∈ [30◦,45◦]). The On/Off normal-
sensitivityofanastroparticletelescopetodetectsources ization factor is assumed to be α = 0.2 in this work.
Aleksi´c et al. (2016) report further cutting in energy
with power law spectra:
toachievethemostsensitiveinstrument. Therefore,I
alsoperformanadditionalcutinenergy,whichtrans-
1. A (E): Effectiveareaasafunctionoftrueenergy
eff lates into a cutoff in the effective area and a reduced
after all cuts background rate.
2. λ : The sensitivity consensus criterion is, unlike References—1) Aleksi´c et al. (2016)
lim
the upper limit criterion, five standard deviations
calculated according to Li & Ma (1983). Subse-
quently required are
The integral spectral exclusion zone method can now
(a) α: On/Off normalization factor beappliedtotheMAGICsensitivityquestion. Whenin-
vestigating the sensitivity, most authors (Aleksi´c et al.
(b) σ : BackgroundeventrateintheOnregion,
bg
2016; Aharonian et al. 2006d; Park 2016) use five stan-
estimated from simulations or independent
dard deviations calculated according to Equation 17
Off region data
in Li & Ma (1983) as sensitivity limit criterion. The
Unfortunately,itisnotcommonforCherenkovtelescope connection between this criterion and the signal counts
collaborationstopublishA (E), α, andσ coherently limit λ can be constructed from assuming a constant
eff bg lim
6 Ahnen
10-8 MAGIC sensitivity backgroundfluxσbg inanareawithOnregionexposure
10-9 N =(σ +σ )t, (12)
on lim bg
10-10 Noff=σbgt/α, (13)
1]−
TeV)10-11 wherethesignalratelimitσlim (tobeinferred)isrelated
2m s 10-12 0.1 h to the signal counts limit via the observation time
N/dE / [(c10-13 MAGIC C.U. 0162...5464. 6hhh h λlim =σlimt. (14)
d ApJ 674 102.4 h
10-14 0.1 C.U
WheninsertingEqn12andEqn13intothelimitcondi-
0.01 C.U
10-15 low Zd tion SLiMa(Non,Noff,α) = 5, one can numerically solve
med Zd for σ . Then, Eqn. 14 can be used to calculate λ .
lim lim
10-11602 103 104 The Li&Ma criterion is only valid for count numbers
E / GeV which are “not too few” (Li & Ma 1983). Therefore, it
Figure 3. ThesensitivityoftheMAGICtelescopetodetect is common (Aleksi´c et al. 2016; Park 2016; Aharonian
power law sources for the analysis parameters from Tab 2, et al. 2006d) to use an additional limit, which is impos-
independentofΓ. Itiscalculatedusingtheintegralspectral
ing to measure at least ten gamma-ray signal counts in
exclusion zone method, for selected observation times t and
the zenith distance ranges low Zd ∈ [0◦,30◦] and medium the On region λlim ≥10.
Zd ∈ [30◦,45◦] as specified in Aleksi´c et al. (2016). Crab In this way, the instrument- and analysis-specific sen-
nebula(Albertetal.2008)likespectra(CrabunitC.U.)are sitivities can be calculated for any observation time t
shown for comparison.
using the integral spectral exclusion zone method.
Table 3. Table of selected northern hemisphere galactic TeV sources
name assoc. tevcat RA dec Zd range f Γ ref. t
0 est
(◦) (◦) (cm2sTeV)−1 (h)
Crab TeV J0534+220 83.629 22.022 low (2.83±0.64)·10−11 −2.62±0.07 1) 0.036+0.010
−0.007
HESS J1837-069 TeV J1837-069 279.410 -6.950 medium (5.0±0.3)·10−12 −2.27±0.06 2) 0.336+0.035∗
−0.031
W 41 TeV J1834-087 278.690 -8.780 medium (3.7±0.6)·10−12 −2.5±0.2 3) 0.489+0.171
−0.124
Cas A TeV J2323+588 350.806 58.808 medium (1.45±0.11)·10−12 −2.75±0.3 4) 2.04+0.61
−0.51
LS 5039 TeV J1826-148 276.563 -14.842 medium (9.1±0.7)·10−13 −2.53±0.07 5) 5.69+0.90∗
−0.72
W 49B TeV J1911+090 287.780 9.087 low (3.2±1.1)·10−13 −3.14±0.34 6) 16+16∗
−7
CTB 87 TeV J2016+372 304.008 37.220 low (3.1±1.5)·10−13 −2.3±0.5 7) 28+58∗
−16
3C 58 TeV J0209+648 31.379 64.841 medium (2.0±1.0)·10−13 −2.4±0.4 8) 100+230
−60
Note—Table showing selected galactic TeV gamma-ray sources and their estimated observation time until detection t .
est
Right ascension and declination are given for the epoch J2000. Calculated values are given in bold. Stars indicate
predictions, as these sources have not been reported by MAGIC.
References—1) Aharonian et al. (2004), 2) Aharonian et al. (2006c), 3) Albert et al. (2006), 4) Kumar (2016),
5) Aharonian et al. (2006a) low state, 6) Abdalla et al. (2016), 7) Aliu et al. (2014b), 8) Aleksi´c et al. (2014)
The result, applied to the MAGIC telescope analyses eightselectedgalacticTeVemittinggamma-raysources,
from Tab. 2, is shown in Fig. 3. It agrees well with observable from the MAGIC site (Latitude 28.76◦N) at
MAGIC’s ability (Aleksi´c et al. 2016) to detect a Crab low or medium zenith distances. This implies source
nebula-like source with 1% Crab nebula flux within ∼ declinations ∈ [−16◦.24,73◦.76]. As the MAGIC back-
25h. ground rate parameters σ from Tab. 2 are valid for
bg
In the case of specific sources, it is useful to return to point-like sources, only MAGIC quasi point-like sources
the power law parameter space {f ,Γ}. Table 3 shows (up to a mean extension of ρ¯ < 0.1◦) are considered.
0
Integral upper limits assuming power law spectra 7
103 SLiMa=5 condition 2.2 t/h until SLiMa=5, source: LS 5039
MAGIC low Zd
MAGIC medium Zd
2.3
102 14.0 7.0
2.4
2.5
t / h 101 Γ
2.6
100 2.7 4.0
2.8
a) b)
10-1
100 101 102 0.6 0.7 0.8 0.9 1.0 1.1 1.2
σlim / h f0 / [(cm2 s TeV)−1] 1e 12
Figure 4. a) The average observation time t until a certain signal rate limit σlim is reached, demonstrated for the MAGIC
analyses shown in Tab 2. b) The average observation time t until MAGIC detects a medium Zd source as a function of the
power law parameters. This is shown in comparison to the measured LS 5039 power law parameter 1σ, 2σ, and 3σ confidence
levels Aharonian et al. (2006a).
In the following, I use the LS 5039 low state flux as an
example for how to estimate the necessary observation
time t until detection.
est
The average time to detect a certain power law in
the power law parameter space must be calculated by
numerically inverting the signal rate limit σ (t). Fig-
lim
ure 4a) shows σ−1 as a function of the signal rate for
lim
the low Zd and medium Zd MAGIC analyses.
Then, the power law limit curve Eqn 4 can be re-
formulated using the average signal counts function λ
s
(Eqn. 3), the power law spectrum dN/dE (Eqn. 1), and
σ−1 as
lim
t=σ−1(c(Γ)f ). (15)
lim 0
Finally, the average estimated observation time t un-
est
til detection (Eqn. 15) can be calculated as a function
Figure 5. Spectra drawn from the parameter space accord-
of the power law parameters {f ,Γ}. This is demon-
0 ingtotheLS5039parametererrorswitharedlineindicating
strated in Fig. 4b) together with the measured LS 5039 the most likely power law. Integral spectral exclusion zones
parameter confidence intervals (Aharonian et al. 2006a) are calculated for three MAGIC times of observation. The
solid line indicates the integral spectral exclusion zone for
assuming uncorrelated normal distributed errors in f
0 the median estimated time to detection t = 5.69h. It
andΓ. ThedataindicatesthatMAGIC,usingtheanal- est
touches the red most likely spectrum at its sensitive energy.
ysis parameters from Tab. 2, would need test ∼4h−7h The dotted lines indicate the evolution of the integral spec-
in order to detect this source. tral exclusion zone, given half or double the median time to
detection.
To quantify t , one can sample from the parameter
est
space according to the LS 5039 parameter errors and
calculate the histogram of times to detection. I report
the times to detection as the median time with asym- Finally, one can illustrate the estimated time to de-
metric errors calculated from the 16th percentile and tectionusingtheintegralspectralexclusionzone, which
the 84th percentile in Tab 3. In the case of LS 5039, is done in Fig. 5. It shows the simple but powerful in-
this results in an estimated MAGIC time to detection terpretation of the integral spectral exclusion zone as
of t = 5.69+0.90h. Four out of eight t calculated theinstrumentdetectioncapability—theborderofthe
est −0.72 est
are predictions as MAGIC has not reported detections region in the spectrum parameter space which any de-
on HESS J1837-069, LS 5039, W 49B, and CTB 87. tectable power law would cross.
8 Ahnen
5. DISCUSSION statistical approaches to choose from. However, the de-
tails of calculating λ are up to each individual exper-
The calculations in this manuscript are done using lim
imentalist and what method she believes in. The signal
the true energy E, as I only consider integral results
count limit λ , formulated as constraint in Eqn 6, is
(see Sec. 3). Therefore, limited energy resolution and lim
only the interface of my integral spectral exclusion zone
migration do not affect the integral spectral exclusion
method with the world of statistics: It stays the same
zone, as a power law source does indeed emit with its
while the statistical methods may be exchanged.
true individual spectrum and an instrument measures
the events according to A (E). This is the case even
eff 6. CONCLUSION
when one is unable to reconstruct the individual event
energies at all. Therefore, it is possible to know the Many high-energy astronomers struggle with the im-
sensitivity of an instrument to detect power law sources plications of non-detections and instrument limits. The
without reconstructing the energy of individual events. universal observation of power laws in high-energy
Thereareseveralcaseswhensourcesrevealadditional sources allows framing these problems in the parameter
curvature, besides the general power law behavior, af- space of the power law model. A simple and powerful
ter measuring them with high significance and low sta- one-to-one relation of the integral spectral limit in the
tistical uncertainty. However, the upper limit question power law parameter space into spectra makes it pos-
and the sensitivity question for an individual telescope sible to construct the integral spectral exclusion zone.
analysis only consider what happens at the detection Thisintegralspectralexclusionzonedemonstrateswhat
threshold and only what happens in the limited energy astroparticletelescopesareactuallycapableofdetecting
domain of A (E) (see Eqn. 3). There, basically ev- – and how long it will take. In this way one can eas-
eff
erysourceinastroparticlephysicsappearsaspowerlaw ilycompareinstrumentperformances, optimizeanalysis
initially. Therefore, I argue that the integral spectral algorithms, and test model predictions of high-energy
exclusion zone is applicable, even when source spectra astrophysics.
deviate “not too much” from power laws. A python package implementing the methods from
InthismanuscriptIonlyconsideredOn/Offmeasure- this manuscript can be found at https://github.com/
ments so far, which infer results in view of Poissonian mahnen/gamma_limits_sensitivity
signal-and-noiseexperiments(seeSec.2),buttheremay
be other cases when, for instance, no background is ex- I would like to thank Adrian Biland, Sebastian A.
pected and λlim can be calculated from a single Poisson Mueller, Scott Lindner, Camila Shirota and Dariusz
process producing Non. In general, there are plenty of Gora for support, discussions, and valuable comments
on the article.
APPENDIX
A. APPLICATION OF THE LAGRANGE MULTIPLIER METHOD FOR CALCULATING THE INTEGRAL
SPECTRAL EXCLUSION ZONE
Motivated by the results shown in Fig. 1, I assume the existence of a maximum flux (Eqn. 1) given the external
constraint (Eqn. 4). Then, the necessary conditions for the existence of the integral spectral exclusion zone can be
deduced from the method of Lagrange multipliers. The first step is the construction of the problem specific Lagrange
functionL(f ,Γ)byjoiningthefunctiontobemaximizeddN/dE totheexternalconstraintusingaLagrangemultiplier
0
δ
(cid:18) E (cid:19)Γ
L(f ,Γ)=f +δ(tf c(Γ)−λ ), (A1)
0 0 E 0 lim
0
where c(Γ) is an abbreviation for the spectrum weighted acceptance
(cid:90) (cid:18) E (cid:19)Γ
c(Γ):= A(E)dE. (A2)
E
0
By taking the gradient with respect to the parameters {f ,Γ} and equating it to zero one gets a set of equations
0
(cid:18) E (cid:19)Γ (cid:18) E (cid:19) ∂c(Γ)
f ln +δtf =0, (A3)
0 E E 0 ∂Γ
0 0
(cid:18) E (cid:19)Γ
+δtc(Γ)=0, (A4)
E
0
Integral upper limits assuming power law spectra 9
which, together with the external constraint Eqn. 4, constitute the necessary conditions. Equation A4 solved for δ
gives
(cid:16) (cid:17)Γ
− E
δ = E0 . (A5)
tc(Γ)
This equation inserted back into Eqn. A3 results in
(cid:18) (cid:19)
E 1 ∂c(Γ)
ln = . (A6)
E c(Γ) ∂Γ
0
When interchanging the integral and differentiation in the explicit formula of ∂c(Γ)/∂Γ and solving Eqn A6 for the
energy E the result is
E =exp(µ(Γ)), (A7)
where µ(Γ) is an abbreviation for the average of the natural logarithm of the energy E over the spectrum weighted
acceptance
(cid:82) EΓA (E)ln(E)dE
eff
µ(Γ):= (cid:82) . (A8)
EΓA (E)dE
eff
µ(Γ) is independent of E , which is a reasonable result for a scale factor. I identify the energy at which the Lagrange
0
function has a maximum with the sensitive Energy E in Sec 3.
sens
Equation A8 shows that the existence of the integral spectral exclusion zone is a direct consequence of the physical
argumentthattherecannotbeinfinitecountswhichleadstothenormalizabilityofthespectrumweightedacceptance
(finite numerator and denominator in Eqn. A8). The average natural logarithm of the energy µ(Γ) is an increasing
functionofΓ,asharderspectraproducehigherenergyeventsmorefrequently. Therefore,E (Γ)isalsoanincreasing
sens
function, which means it is invertible. Finally, one can take Eqn. A7, solve it numerically for Γ, and find the last
remaining undetermined parameter f by solving the so far unused constraint function Eqn. 4, which is shown in
0
Eqn. 9.
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