Inferred cosmic-ray spectrum from Fermi-Lat -ray observations of the Earth’s limb
Recent accurate measurements of cosmic-ray (CR) species by ATIC-2, CREAM, and PAMELA reveal an unexpected hardening in the proton and He spectra above a few hundred GeV, a gradual softening of the spectra just below a few hundred GeV, and a harder spectrum of He compared to that of protons. These newly-discovered features may offer a clue to the origin of high-energy CRs. We use the Fermi Large Area Telescope observations of the -ray emission from the Earth’s limb for an indirect measurement of the local spectrum of CR protons in the energy range GeV–6 TeV (derived from a photon energy range 15 GeV–1 TeV). Our analysis shows that single power law and broken power law spectra fit the data equally well and yield a proton spectrum with index and above GeV, respectively.
pacs:96.50.sb, 95.85.Ry, 98.70.Sa
Introduction. The spectrum of CRs has offered few clues to its origin so far. The generally accepted features are at very-high and ultra-high energies (see, e.g., Figure 1 in Swordy (2001)): the so-called “knee” at a few thousand TeV Kulikov and Khristiansen (1958); Haungs et al. (2003), the second “knee” at TeV, the “ankle” at higher energies Abbasi et al. (2005), and a spectral steepening above TeV Abbasi et al. (2009); Abraham et al. (2010). It is believed that CRs below the second knee are Galactic, while extragalactic CRs dominate at higher energies (Meister, 1991; Strong et al., 2007).
The data recently collected by three experiments, ATIC-2 Wefel et al. (2008); Panov et al. (2009), CREAM Ahn et al. (2010); Yoon et al. (2011), and PAMELA Adriani et al. (2011), indicate a new feature at relatively low energy: a break (or hardening) of CR proton and He spectra at GV in rigidity. PAMELA claims to detect the break at 95% confidence level for both species. Below the break, PAMELA data agree very well with the earlier data from AMS-01 Alcaraz et al. (2000) and BESS Haino et al. (2004). Above the break, ATIC-2 results agree well with those of CREAM, smoothly connecting to highest energy points from PAMELA. The change in the spectral indices for both protons and He is .
However, the break itself is observed only by PAMELA near its high-energy limit. Much evidence of this newly discovered break or flattening comes from a combination of data by several different experiments, which may be subject to cross-calibration errors. A verification of this new feature requires an independent confirmation, preferably with a single instrument. Meanwhile, recent preliminary AMS-02 results111http://www.ams02.org/wp-content/uploads/2013/07/Proton_2.jpg do not show any feature in the proton and He spectra up to TeV and also seem to contradict ATIC-2 and CREAM results. In this paper, we demonstrate that such a measurement can also be done indirectly through observation of the CR-induced -ray emission from the Earth’s atmosphere.
Atmospheric -ray emission is mainly the result of hadronic CR cascades: CRs entering the atmosphere near grazing incidence produce showers that develop in the forward direction, resulting in a very bright -ray signal from the Earth’s limb as seen from orbit. The -ray spectrum from CR interactions at the very top of the atmosphere depends only on the inclusive -ray production cross section and the spectrum of CR particles. If the cross section is known, the shape of the local CR spectrum can be recovered from the -ray spectrum. However, this method can only measure the total spectrum of CRs. To deduce the spectrum of protons, the most abundant component of CRs, one has to assume a spectrum of He, the second most abundant component. The contribution of the latter to the total -ray emission is –20%, depending on the energy. Therefore, accurate modeling of the contribution from He interactions is not very critical, and heavier nuclei can be neglected.
Observations of Galactic diffuse -ray emission (GDE) have provided valuable information about CR spectra in distant locations Strong et al. (2000, 2004); Porter et al. (2008); Abdo et al. (2009a); Ackermann et al. (2012a) and in the local interstellar medium Abdo et al. (2009b); Abdo et al. (2010). Similarly, observations of the Earth’s limb -ray emission can be used to deduce the CR spectrum near the Earth. In contrast with the GDE, the contribution from the inverse Compton scattering of CR electrons to the Earth’s limb emission is negligible. Furthermore, viewed from low-Earth orbit, the limb is orders of magnitude brighter than the GDE. Using the Earth’s emission is, therefore, a simpler way to derive the spectrum of CR nucleons than using the GDE.
The -ray emission from the Earth’s limb was first observed by the SAS-2 Thompson et al. (1981) and EGRET Petry (2005) instruments, but these observations were limited in statistics and angular resolution. Fermi-LAT made the first measurement of the Earth’s limb -ray emission above 10 GeV Abdo et al. (2009c) and was able to resolve the limb profile to discriminate between the thin and thick target regimes, demonstrating its capability for indirect measurements of the CR spectrum. In this paper we report on the analysis of 5 years of Earth’s limb observations with the Fermi-LAT.
Data and Analysis Method. Fermi was launched in June 2008 and spent the first few weeks calibrating the instruments during the Launch and Early Operations (L&EO) period, during which the LAT was tracking a few well-known bright sources, allowing the Earth’s limb to frequently enter the field of view (FoV). In September 2008, three hours of limb-stare observations were performed. This is the data set used in Abdo et al. (2009c) and part of the data set in the analysis presented here. The additional part of the data set is described as follows.
The spacecraft operates mainly in survey mode, keeping the Earth’s limb far from its boresight as it is a background for other analyses. However, for a small fraction of the operating time, the LAT performs pointed observations by following a celestial target while it is not occulted by the Earth, including while it is near the limb. We select this pointed data set by accepting events when the magnitude of the rocking angle222The angle between the LAT’s boresight and the zenith is called the rocking angle. is , greater than that for the normal survey mode, up to August 8, 2013. This rocking angle selection rejects the survey mode data, for which the Earth’s limb photons have large () incidence angle.
|Observation type||Start date||End date||Livetime (days)||N|
|L&EO||Jul 15, 2008||Jul 30, 2008||9||967|
|Limb-stare||Sep 29, 2008||Sep 29, 2008||0.125||18|
|Pointed (multiple)||Aug 21, 2008||Aug 8, 2013||90||6762|
We avoid the geomagnetic and solar modulation of local CRs near the Earth by considering only rays above 15 GeV because they must be produced by CR protons with energies of at least (but mostly much greater than) 15 GeV. The resulting number of Earth’s limb photons above 15 GeV (N) and the average incidence angles measured from the LAT’s boresight () for these datasets are in Table 1.
The data are analyzed here in the local nadir coordinates, in which is the angle measured from the nadir direction at the location of the LAT. At the LAT’s altitude of km, the physical limb of the Earth is at . However, the peak of the -ray emission above 15 GeV is at , due to the height of the atmosphere and the effects of -ray absorption as discussed in detail in Abdo et al. (2009c). At , the integrated column density for grazing-incidence particles is g cm (see Figure 5 in Abdo et al. (2009c)). From this angle outwards, the atmosphere is in the thin-target regime with photons produced from a single interaction, the absorption effects are negligible, and the resulting -ray spectrum is determined by the local spectrum of CRs. Thus, is the inner edge for the studies presented here. The outer edge of the Earth’s limb is chosen as because the emission from celestial sources starts to dominate for larger angles.
We use the P7REP reprocessed data (see Bregeon et al. (2013) and FSSC page333http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Pass7REP_usage.html for details) with the P7REP_SOURCE event selection and the associated P7REP_SOURCE_V15 instrument response functions. We apply two additional cuts: (reduced incidence angle) to avoid the edge of the FoV, which is prone to systematic uncertainties, and (thin-target regime) to select photons from the Earth’s limb.
The background is estimated from a ring surrounding the Earth’s limb (). The ring immediately surrounding the limb was not used in order to avoid spill-over photons from the limb due to the LAT’s point-spread function (PSF). The background level shown in Figure 1 is small, ranging from % at 15 GeV to % at 500 GeV of the bright limb emission.
For the P7REP data used here, dedicated simulations and flight data comparisons have been performed to validate the LAT responses up to 1 TeV by the LAT Collaboration. Based on these studies, we adopt an effective area () uncertainty of 5% at 10 GeV, increasing linearly with the logarithm of energy to 15% at 1 TeV444 http://fermi.gsfc.nasa.gov/ssc/data/analysis/LAT_caveats.html.
We simulate many realizations of the energy dependence of the which obey the above estimated uncertainty to observe the effect of instrumental systematic error. Specifically, to obtain one realization, we generate 3 random numbers at 10, 100, and 1000 GeV from Gaussian distributions for which the mean is 0 and is the value of the uncertainty estimation at the three energy points. Cubic spline interpolation between these points describes the deviation of from the central value, which would then distort the measured Earth’s limb spectrum in a way consistent with the systematic uncertainties, allowing us to evaluate the propagated uncertainties of the final results. This algorithm to simulate uncertainties assumes uncorrelated errors for two energy bins that are sufficiently far apart (larger than half a decade in energy), but the interpolation results in highly correlated errors between nearby energy bins (see Section 5.6.2 in Ackermann et al. (2012b)).
We also correct for the angular resolution effects on the limb spectrum itself. The dominant effect is contamination by limb photons from , where the emission is brighter. The other is the leakage of photons from the limb to each side of the defined boundary. Above 15 GeV, where the LAT’s PSF is narrow (68% containment at ) and not strongly energy dependent (see Figure 57 in Ackermann et al. (2012b)), these corrections combined decrease the measured intensities by %, depending on energy. The effect on the spectral index is relatively small compared to that from .
We determined that +2%/-5% uncertainty of the absolute energy scale (described in Section 7.3.4 in Ackermann et al. (2012b)) translates into % effects on the absolute normalization of the spectrum, which does not alter the results presented here.
To infer the CR proton spectrum from the -ray measurement, we use two interaction models, one by Kamae et al. (2006) (Kamae model) and the other by Kachelrieß and Ostapchenko (2012) (K&O model). For each model, we calculate the -ray spectrum by integrating a model of the proton spectrum from GeV to TeV in kinetic energy.
In our study, we assume that the atmosphere consists of 100% Nitrogen. This does not affect our results because studies of proton-nucleus interactions at high energies (e.g., Orth and Buffington (1976); Atwater and Freier (1986)) show that the cross section can be scaled from the cross section by applying an energy-independent scaling factor , where is the atomic number of the nucleus. The precise scaling factor for the atmospheric composition is not important for this analysis because it changes only the normalization of the fitted proton spectrum.
Observations of the -ray emission from the Earth’s limb cannot discriminate between contributions of CR protons and heavier nuclei. Thus, we must rely on the direct measurements of the CR composition. The He fraction in CRs is about 6–10% by number, depending on energy, in our energy range of interest, so its contribution has to be taken into account, while the contribution of the heavier nuclei can be safely neglected. There are a number of empirical parameterizations for nucleus-nucleus meson multiplicity (e.g., Appendix A in Orth and Buffington (1976) and Eq. (3b) in Atwater and Freier (1986)). These formulas give similar values for the ratio of to cross sections, /. We use this number to scale the -interaction models and to calculate the relative contribution of He nuclei to the limb -ray emission. Since the contribution of rays produced by He is –20% depending on energy, and most of the emission is produced by protons, the scaling uncertainty has little influence on the final fit results.
To determine the He spectrum, we fit the combined PAMELA (Adriani et al. (2011)), CREAM (Yoon et al. (2011)), and ATIC-2 (Panov et al. (2009)) He data above 50 GeV/n (102 GV) with spectral forms described below Equation 1. According to PAMELA measurements Adriani et al. (2011), the He/ ratio at GeV is 6.2%. We use this value together with the cross section scaling to fix the contribution of He to photon production at 15 GeV to 9.5%. We then forward-fold by varying the parameters of the input proton spectrum so that the resulting -ray spectrum calculated from the -interaction models provides the best fit to the Earth’s limb measurement. In the fitting procedure, the normal Poisson likelihood function is maximized:
where is the number of energy bins. is the Poisson probability of observing counts given that the model predicts counts (-ray model flux exposure) for the energy bin. We use two models for the local CR proton and He spectra in the fitting procedure:
SPL: Single power law in rigidity. This model assumes a single power law for CR protons and He. For He, the index is our best-fit value of the combined PAMELA, CREAM, and ATIC-2 data. For protons, we fit both the normalization and index to our measurement of the -ray spectrum from the Earth’s limb.
BPL: Broken power law in rigidity. This model assumes a broken power law for both proton and He spectra. As before, the He spectrum is fixed to the best fit of the combined direct measurements, for which the spectral index changes from to at GV. We then fit the indices, break energy (), and normalization for the proton spectrum.
We evaluate the statistical uncertainties of the fit results by fitting a large number of simulated realizations of photon counts generated with a Poisson distribution for which the expected value is the measured count in each energy bin. Likewise, simultaneous simulations of photon counts and ranges of , as previously described, give the total (combined systematic and statistical) errors. We also add in quadrature the 5% absolute energy scale uncertainty to the errors of the fitted energy parameter.
Results. The measured -ray thin-target limb spectrum, the background-sky flux (which has already been subtracted from the limb spectrum), and the best-fit -ray models are shown in Figure 1. The -ray emission from the local H i555Work in preparation by the LAT Collaboration is scaled to approximately match that from the Earth’s limb and shown for comparison. As expected, the two agree well above GeV, since they are produced from hadronic interactions by the same local population of CRs in the thin-target regime. Below GeV, the spectra differ due to the geomagnetic and solar modulations of CRs in the vicinity of the Earth, reducing the number of CRs interacting with the Earth’s atmosphere. For this reason we limited our study to limb rays above 15 GeV. The approximate proton-to--ray energy conversion factor for the power-law spectrum of protons is 0.17 Kelner et al. (2006). The energy range of the inferred proton spectrum is thus GeV–6 TeV.
|BPL index 1|
|BPL index 2|
|BPL vs SPL|
Using the K&O and Kamae models, we obtain the results shown in Table 2. The log likelihood for the best-fit BPL is better than that for the best-fit SPL. To account for systematic uncertainties, we apply Monte Carlo simulations to translate this likelihood ratio into a significance. By assuming that the best-fit SPL is the true underlying flux model, we produce simulations of from the estimated errors as previously discussed, for each of which we generate simulations of SPL realizations. We then fit the distribution of the log likelihood differences between SPL and BPL for these M total simulations with a Gaussian function and evaluate how likely it is that the best-fit BPL we obtain from the actual measurement would give a log likelihood difference of 0.9 or above as compared to the SPL. We find that it corresponds to a significance of .
We performed several cross checks to test the stability and consistency of the results. We studied the effects of using the event selection with more stringent rejection of residual CRs (P7REP_CLEAN), tighter incidence angle () cuts, and reasonable variations of the fitted energy ranges (up to 20 GeV lower bound and down to 120 GeV upper bound in -ray energy). All of these cases yield consistent results.
Figure 2 shows the resulting best-fit SPL and BPL derived from the K&O model in comparison with direct measurements, assuming an effective atmospheric column density of g cm, as described below.
In order to determine the absolute normalization of the inferred proton spectrum, we use the NRLMSISE-00 atmospheric model Picone et al. (2002) to calculate the average line-of-sight column density, weighted by -ray intensity, in the range studied here (–) to be 1.2 g cm. Due to the exponential change of the atmospheric density with , the evaluated density is extremely sensitive to the lower bound of the range. We thus empirically adjust the absolute normalization of our inferred proton spectrum to approximately match that of direct measurements as shown in Figure 2 by changing the atmospheric column density from 1.2 g cm to 1.0 g cm. This is equivalent to increasing the lower bound of from 68.40 to 68.42 when we calculate the atmospheric column density. The small change in the effective lower bound of by has many potential justifications, such as the LAT altitude variations which smear the precise calculation of the target density, the atmospheric model uncertainties, and other absolute normalization uncertainties as previously discussed. Since our primary interest is in the spectral indices, the difference in normalization is of no importance.
Discussion and Conclusion. Our LAT analysis, which employs a different technique from direct measurements, shows that the CR proton spectrum between GeV–6 TeV can be described equally well () with the SPL and BPL models. The best-fit spectral indices ( for SPL and above GeV for BPL) are consistent with each other.
We note that our best-fit SPL index is from the value () reported by PAMELA for a lower energy range (29–79 GeV). However, our best-fit SPL index for GeV–6 TeV is in good agreement with the fitted index for GeV–1 TeV reported by PAMELA Adriani et al. (2011) and with the measurements at higher energies by ATIC-2 Panov et al. (2009) and CREAM Yoon et al. (2011). While Fermi-LAT results cannot confirm or disprove the existence of the spectral break itself yet, they do indicate a flatter proton spectrum at high energies, consistent with direct measurements by ATIC-2 and CREAM.
This result is the first indirect measurement of the proton spectrum in the energy range GeV–6 TeV using observations of the -ray emission of the Earth’s limb. Continuing observations with Fermi-LAT will allow us to improve the precision of the measurement of the CR spectrum and extend the energy range.
The Fermi-LAT Collaboration acknowledges support from a number of agencies and institutes for both development and the operation of the LAT as well as scientific data analysis. These include NASA and DOE in the United States, CEA/Irfu and IN2P3/CNRS in France, ASI and INFN in Italy, MEXT, KEK, and JAXA in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the National Space Board in Sweden. Additional support from INAF in Italy and CNES in France for science analysis during the operations phase is gratefully acknowledged. I.V.M. acknowledges support from NASA grants NNX11AQ06G and NNX13AC47G.
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