Investigation on the cosmic-ray shadow of planets and asteroids

The moon shadow and sun shadow of cosmic rays are commonly used to calibrate the angular resolution of the instrument in extensive air shower experiments, measure the proton-antiproton ratio, and study the interplanetary magnetic field (IMF). The shadow effect of planets and asteroids in the solar system, on the other hand, has received little attention. If considerable shadow effects can be observed, a novel approach may be developed to calibrate the point spread function and investigate the IMF. In this work, we calculate the sensitivity of observing the shadow effects of planets and asteroids in the next hundred years using LHAASO’s instrumental response as an example. The result shows that the blocking impact of these celestial bodies is minimal; thus, their influence on the direction distribution of cosmic rays is negligible.


Introduction
Cosmic rays (CRs) are blocked by some celestial objects when propagating, resulting in a shadow in the sky. We call these phenomena the "shadow effect" of the celestial objects. The moon shadow [1][2][3][4] and the sun shadow [5,6] are representatives of these phenomena. People have been studying the moon shadow and the sun shadow for a long time. As early as 1957, G.W.Clark noted the shadow effect of the moon while investigating the arrival directions of air showers [7]. Many ground-based detectors have observed moon shadow and sun shadow in the past decades, including H.E.S.S. [8], ARGO-YBJ [9], HAWC [10] and LHAASO [11].
Extensive air shower(EAS) experiments usually use the circular banded distribution of the moon shadow to determine the angular resolution [12] because the apparent radius of the moon is about 0.26 • , which is smaller than a typical angular resolution of this kind of experiment. For example, LHAASO-WCDA has an angular resolution of ∼0.45 • at a few TeV [13]. As charged cosmic rays are deflected by the geomagnetic field(GMF), the deflection angle increases with the decrease of particle's energy. Thus, the EAS experiments use moon shadow to determine the detector's energy resolution [4] and study the ratio of proton and antiproton [14]. Similarly, sun shadow can be used as a calibration source for a neutrino telescope [15] and measure the interplanetary magnetic field (IMF) [16].
In addition, other celestial objects should have similar "shadow effects." However, all previous research has not discussed them seriously. In this study, we analyze the shadow effects of the planets and asteroids in the solar system in this work. These sources are more like point sources, and we can use their shadow to calibrate the point spread function (PSF) of high-energy cosmic rays. It could be helpful to study IMF at lower energies. The apparent radii of these objects are relatively small, so we stack the shadow effects of these objects. With the construction and operation of modern large-scale ground-based EAS experiments, it is time to investigate the possibility of detecting the shadows of these objects.
About 120,000 asteroids have been observed; most are located in an asteroid belt between Jupiter and Mars. However, only just over 20,000 asteroids with their orbit properties were accurately recoded by available databases. This paper will use these 20000 asteroids to estimate the sensitivity of observing these objects' cosmic-ray shadow effect, taking LHAASO's instrumental response. Besides the analysis for individual objects, a stacked analysis is also performed.

Methods of analysis
We use the likelihood ratio method to estimate the significance level of the cosmic ray shadow of a celestial body. The test statistic is defined as with the background-only hypothesis H 0 and the background-plus-signal hypothesis H 1 . L is the likelihood. N obs means observed CR events. N s is the number of blocked CR events by a specific object within the object-centered round region, whose radius is defined as the smooth radius of r sm ; N b is the background events(average event numbers within the same size of regions excluding sources) estimated by the background estimation method, such as the direct integral method [17] and the equizenith angle method [18]. The radius of the round region depends on the PSF of the detector in the point source case. If the PSF can be defined as a single Gaussian distribution,the optimized radius is r sm =1.51 gauss [13] for the best significance level when we use the top-hat smooth method [19].
In the background only case, the TS value follows a 2 distribution with n degrees of freedom according to the Wilks theorem [20]. For a single celestial body, the significance is S= √ TS in the case of one free parameter case. Meanwhile, as the EAS experiment will collect a large number of events, the N b and N obs follow a Gaussian distribution. The expected significance can be approximated by is small according to the moon shadow analysis due to the limited angular resolution of a EAS experiment. The significance can be expressed as N s and N b depend on the zenith angle , which can be estimated by where r 0 is the apparent radius of a given celestial body, and r sm is the smooth radius as defined above. (t) is the collected CR event density in the zenith direction at time t. The detection efficiency depends on the atmospheric depth the cosmic ray passes through and the effective area for collecting, thus related to the zenith angle's cosine value. We use cos n ( ) to estimate the CR detection efficiency as a function of the zenith angle, and we obtain n = 7 from a simulation by Corsika [21]. The event density is (t)cos n ( (t)) at the given zenith angle .
In general, r 0 is much less than the instrument's angular resolution, so the r sm is mainly determined by the angular resolution. In our case, the sigma is about 0.56 • , and r sm is about 0.84 • [13]. We used the same r sm for planets, asteroids and the moon in this work, as all these celestial bodies can be considered as point sources. sm is the collection coefficient at a radius of r sm . In the case of r sm = 1.51 gauss regarding a Gaussian PSF, 68% of the signals are included, i.e., sm =0.68.
As the variation of (t) is negligible, we assume that (t) is stable. The significance now is We define the variable = ∫ t r 2 o (t)cos n ( (t))dt √ ∫ t cos n ( (t))dt to estimate the relative significance for each celestial body. We define R = ∕ moon , R Ns = Ns∕Ns moon to represent the significance and blocked event counts with respect to that of the moon shadow, respectively. Note that the expected significance and blocked event counts are proportional to the apparent radius square. To perform a stacked analysis, we define a stacked test statistics as [22]: For every celestial body, we can estimate its significance with S i = √ TS i following the same test procedure; therefore, we obtain the expected stacked significance where S i is significance of the ith celestial body. We end up with where the stack , i denote stacked one and ith one, respectively.

Planets and asteroids
We calculated the of every planet and stack of asteroids in the solar system. The trajectory data are set from January 1st, 2020 to December 30th, 2119, obtained from PyEphem [23]. We take the LHAASO experiment as an example and set the site at 90.52 • E , 30.11 • N [24]. Figure 1 shows the residence times of the moon and major planets at different zenith angles. The zenith angles are restricted to less than 50 • because the detection efficiency and the effective area of very inclined events will worsen. Table 1 shows the apparent radius and calculated for the other seven planets. Jupiter and Venus, the top two planets with the largest apparent radius, are the dominant contributors. The mean apparent radius of Jupiter is 0.0053 • , while that is ∼ 0.26 • for the Moon. The apparent area ratio is 4.12 × 10 −4 , while the R is 4.19 × 10 −4 . It indicates that the effect from zenith angle distribution is minimal. We combine all the planets and obtain the stacked R ,stack of 4.6 × 10 −4 using formula (7), as shown in Table 1.
More than 120000 asteroids have been discovered by 2006 and it is estimated that there would be more than one million asteroids in the solar system. These asteroids are distributed in the "asteroids belt" between Mars and Jupiter. These asteroids are approximately 2.2 to 3.6 AU away from the sun, and their revolution periods are typical 3.5 to 6 years.
All observable asteroids analysed in this paper are divided into three groups according to the standards of the Ephem database [23], which is Critical − list Fig. 1 The residence times of the moon and major planets at different zenith angles with from 2020 to 2119. The solid blue line indicates zenith angles' residence times for the moon, and the orange and brown dashed lines are for Jupiter and Venus, respectively. Here the zenith angles are limited to less than 50 • Table 1 The blocking effect of cosmic rays by primary planets in the solar system Dis sun is the average distance between the planet and the sun. Dis earth is the average distance between the planet and the earth. Radius is the average apparent radius of the planet

Numbered Minor Planets(CNMPs), Distant Minor Planets(DMPs) and Unusual
Minor Planets(UMPs). Figure 2 shows the apparent radii distributions of each group of asteroids. The average apparent radii for each group are listed in Table 2. The UMPs group dominates the contribution, the R of 5.31 × 10 −5 .

Result and discussion
Except Neptune, the maximum revolution period of the planets in the solar system is 84 years. To ensure that we could analyze the objects with more than one revolution period, we estimated the shadow effect by using the predicted position of planets and asteroids in the next 100 years. The blocked events ratio R Ns,stack of all celestial bodies is 1.16 × 10 −3 as shown in Table 2, and combining Table 1 we can know that the UMPs and Jupiter make up the majority. The time evolution of over 100 years is shown in Fig. 3. The red solid line indicates the stacked for all planets and asteroids in 100 years. The black dashed line is the 5 × 10 −4 moon , which is served as a reference. The stacked value of R is about 4.61 × 10 −4 for all the planets and asteroids. To get a shadow effect with 5 , it will take 15685 years for KM2A and 10892 years for WCDA. For comparison, the significance of moon shadow effect would be 10846 by this time (25 per month for LHAASO-KM2A [25] at ∼ 20 TeV and 30 per month for LHAASO-WCDA [25] at ∼ 1 TeV).
The blocking effect of these celestial bodies is mainly related to the square of their apparent radii. The stacked blocking area is much less than that of the moon, Fig. 2 The apparent radius distribution of the asteroids in the solar system(Asteroids whose apparent radii are 0 • are not included.). According to [23], there are three groups of asteroids in the solar system. The three subgraphs are the apparent radius distributions of CNMPs, DMPs and UMPs, respectively. so the influence on the direction of cosmic rays caused by these celestial bodies' shadow effect is negligible. The total five significance of all these objects is expected to be observed by LHAASO in a very long time. However, future experiments will give actual observations with the continuous improvement of observation technology, e.g., improved angular resolution and effective area. If the effective area could be increased by an order of magnitude (the detector area of WCDA is 78000m 2 ) and the angular resolution could be increased to 0.05 • , we can observe such an effect within 10 years. If the PSF is much less than the moon's apparent radius, the cosmic ray shadow of planets and asteroids discussed in this work can be served as point sources to calibrate the angular resolution for high-energy cosmic rays.
Furthermore, studying the planets' shadows could play an essential role in analyzing GMF and IMF. The relationship between IMF and heliocentric distance is given in [26]. Previously, the relationship between IMF and heliocentric distance based on the variation of sun shadow has been analyzed [26]. If the planets' shadows are significant enough, we can use them to test models about IMF. The planets' shadow effect provides a variety of methodological choices for the analysis of GMF and IMF because the distances between the planets and the Earth are diverse (ranging from 1.04AU to 30.02AU, as shown in Table 1).