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As the number of low Earth orbit (LEO) satellites rapidly increases, the consideration of frequency sharing or cooperation between geosynchronous Earth orbit (GEO) and LEO satellites is gaining attention. In this paper, we consider a hybrid GEO-LEO satellite network where GEO and LEO satellites are distributed according to independent Poisson point processes (PPPs) and share the same frequency resources. Based on the properties of PPPs, we first analyze satellite-visible probabilities, distance distributions, and association probabilities. Then, we derive an analytical expression for the network's coverage probability. Through Monte Carlo simulations, we verify the analytical results and demonstrate the impact of system parameters on coverage performance. The analytical results effectively estimate the coverage performance in scenarios where GEO and LEO satellites cooperate or share the same resource., Comment: 5 pages, 4 figures, 1 table, submitted to IEEE Transactions on Vehicular Technology

We study the phenomenology of the charged Higgs boson, $H^\pm$,appearing in the fermionic dark matter model mediated by the dark $Z$ boson. This model is in favor of the light dark $Z$ boson, $Z'$, and the light additional neutral Higgs boson, $h$. We find that $H^\pm \to W^\pm h$ and the $H^\pm \to W^\pm Z'$ are dominant decay channels. Thus the promising final states are trilepton signals, $e \mu \mu$ or $\mu \mu \mu$ following $Z' \to \mu^+ \mu^-$ decays and leptonic decays of the $W^\pm$ boson. The charged Higgs boson will be produced from the top quark decays $t \to b H^\pm$ following $t \bar{t}$ production, if $H^\pm$ is light. Whereas $H^\pm$ is heavier than the top quark, the dominant production processes are associated productions with either $Z'$ or $h$, $pp \to W^\star \to H^\pm h$ and $pp \to W^\star \to H^\pm Z'$. We explore the discovery potential of the charged Higgs boson at the LHC. We also discuss the implications of dark matter in relation with the charged Higgs phenomenology., Comment: 12 pages, 4 figures

Due to the high altitudes and large beam sizes of satellites, the curvature of the Earth's surface can impact system-level performance. To consider this, 3GPP introduces the UV-plane beam mapping for system-level simulations of non-terrestrial networks (NTNs). This paper aims to provide a comprehensive understanding of how beams and user equipments (UEs) are placed on the UV-plane and subsequently mapped to the Earth's surface. We present a general process of projecting UEs on the UV-plane onto the Earth's surface. This process could offer a useful guideline for beam and UE deployment when evaluating the system-level performance of NTNs., Comment: 5 pages, 9 figures, 1 table

Low Earth orbit (LEO) satellite communications (SATCOM) with ubiquitous global connectivity is deemed a pivotal catalyst in advancing wireless communication systems for 5G and beyond. LEO SATCOM excels in delivering versatile information services across expansive areas, facilitating both unicast and multicast transmissions via high-speed broadband capability. Nonetheless, given the broadband coverage of LEO SATCOM, traffic demand distribution within the service area is non-uniform, and the time/frequency/power resources available at LEO satellites remain significantly limited. Motivated by these challenges, we propose a rate-matching framework for non-orthogonal unicast and multicast (NOUM) transmission. Our approach aims to minimize the difference between offered rates and traffic demands for both unicast and multicast messages. By multiplexing unicast and multicast transmissions over the same radio resource, rate-splitting multiple access (RSMA) is employed to manage interference between unicast and multicast streams, as well as inter-user interference under imperfect channel state information at the LEO satellite. To address the formulated problems non-smoothness and non-convexity, the common rate is approximated using the LogSumExp technique. Thereafter, we represent the common rate portion as the ratio of the approximated function, converting the problem into an unconstrained form. A generalized power iteration (GPI)-based algorithm, coined GPI-RS-NOUM, is proposed upon this reformulation. Through comprehensive numerical analysis across diverse simulation setups, we demonstrate that the proposed framework outperforms various benchmarks for LEO SATCOM with uneven traffic demands., Comment: 39 pages, 9 figures

Satellite networks are emerging as vital solutions for global connectivity beyond 5G. As companies such as SpaceX, OneWeb, and Amazon are poised to launch a large number of satellites in low Earth orbit, the heightened inter-satellite interference caused by mega-constellations has become a significant concern. To address this challenge, recent works have introduced the concept of satellite cluster networks where multiple satellites in a cluster collaborate to enhance the network performance. In order to investigate the performance of these networks, we propose mathematical analyses by modeling the locations of satellites and users using Poisson point processes, building on the success of stochastic geometry-based analyses for satellite networks. In particular, we suggest the lower and upper bounds of the coverage probability as functions of the system parameters, including satellite density, satellite altitude, satellite cluster area, path loss exponent, and Nakagami parameter $m$. We validate the analytical expressions by comparing them with simulation results. Our analyses can be used to design reliable satellite cluster networks by effectively estimating the impact of system parameters on the coverage performance., Comment: submitted to IEEE Transactions on Communications

We perform global fits of the Higgs boson couplings to the full Higgs datasets collected at the LHC with the integrated luminosities per experiment of approximately 5/fb at 7 TeV, 20/fb at 8 TeV, and up to 139/fb at 13 TeV. Our combined analysis based on the experimental signal strengths used in this work and the theoretical ones elaborated for our analysis reliably reproduce the results in the literature. We reveal that the LHC Higgs precision data are no longer best described by the SM Higgs boson taking account of extensive and comprehensive CP-conserving and CP-violating scenarios found in several well-motivated models beyond the SM. Especially, in most of the fits considered in this work, we observe that the best-fitted values of the normalized Yukawa couplings are about $2\sigma$ below the corresponding SM ones with the $1\sigma$ errors of 3%-5%. On the other hand, the gauge-Higgs couplings are consistent with the SM with the $1\sigma$ errors of 2%-3%. Incidentally, the reduced Yukawa couplings help to explain the excess of the $H\to Z\gamma$ signal strength of $2.2\pm 0.7$ recently reported by the ATLAS and CMS collaborations., Comment: 44 pages, 14 figures, 23 tables; To appear in PRD, $H \to Z\gamma$ considered more rigorously: a few typos removed

The extensive coverage offered by satellites makes them effective in enhancing service continuity for users on dynamic airborne and maritime platforms, such as airplanes and ships. In particular, geosynchronous Earth orbit (GEO) satellites ensure stable connectivity for terrestrial users due to their stationary characteristics when observed from Earth. This paper introduces a novel approach to model and analyze GEO satellite networks using stochastic geometry. We model the distribution of GEO satellites in the geostationary orbit according to a binomial point process (BPP) and examine satellite visibility depending on the terminal's latitude. Then, we identify potential distribution cases for GEO satellites and derive case probabilities based on the properties of the BPP. We also obtain the distance distributions between the terminal and GEO satellites and derive the coverage probability of the network. We further approximate the derived expressions using the Poisson limit theorem. Monte Carlo simulations are performed to validate the analytical findings, demonstrating a strong alignment between the analyses and simulations. The simplified analytical results can be used to estimate the coverage performance of GEO satellite networks by effectively modeling the positions of GEO satellites., Comment: 12 pages, 9 figures, submitted to IEEE Transactions on Wireless Communications

The mediator particle between the Standard Model sector and a hidden sector might have a long lifetime to show the observable displaced vertices in experiments. Considering a fermionic dark matter model in which the hidden sector is connected to the Standard Model by an additional Higgs doublet field, the mediator dark $Z$ boson may live long enough. We explore the possibility to observe the displaced vertices of the long-lived dark $Z$ boson at the CERN LHC and at the proposed SHiP experiment. We find that the ATLAS and CMS searches for the long-lived dark $Z$ boson can probe the mass range $7 < m_{Z'} < 150~{\rm MeV}$ with 150 fb$^{-1}$ integrated luminosity at the LHC run 3, and the SHiP experiment will probe $2 m_e < m_{Z'} < 15~{\rm MeV}$ range with $6 \times 10^{20}$ protons on target in total 15 years. The dark matter phenomenology is also discussed in the region where such a long-lived mediator is detectable., Comment: 11 pages, 4 figures

The nearby radio galaxy M87 offers a unique opportunity to explore the connections between the central supermassive black hole and relativistic jets. Previous studies of the inner region of M87 revealed a wide opening angle for the jet originating near the black hole. The Event Horizon Telescope resolved the central radio source and found an asymmetric ring structure consistent with expectations from General Relativity. With a baseline of 17 years of observations, there was a shift in the jet's transverse position, possibly arising from an eight to ten-year quasi-periodicity. However, the origin of this sideways shift remains unclear. Here we report an analysis of radio observations over 22 years that suggests a period of about 11 years in the position angle variation of the jet. We infer that we are seeing a spinning black hole that induces the Lense-Thirring precession of a misaligned accretion disk. Similar jet precession may commonly occur in other active galactic nuclei but has been challenging to detect owing to the small magnitude and long period of the variation., Comment: 41 pages, 7 figures, 7 tables

We study the kinetic mixing between the cosmic microwave background (CMB) photon and the birefringent dark photon. These birefringent dark photon may exist in parity-violating dark sector, for example, through the coupling to axion field. We show that the birefringence of the dark photon propagates to the CMB photon, but the resulting birefringence may not be isotropic over the sky, but will be anisotropic in general. Moreover, our investigation sheds light on the essential role played by kinetic mixing in the generation of two fundamental characteristics of the CMB: circular polarization and spectral distortion., Comment: 14 pages, 2 figures, 1 table. v2: published version

We study constraints on the hidden sector model mediated by an additional SU(2) Higgs doublet from the phenomenology of Higgs bosons. The hidden sector is assumed to contain a hidden U(1) gauge symmetry and the hidden U(1) gauge boson gets the mass by the electroweak symmetry breaking to be a dark Z boson. The Higgs sector of the model is similar to that of the two Higgs doublet model of type I except for the absence of the CP-odd scalar boson. Using the programs of HiggsBounds and HiggsSignals, we incorporate current experimental limits from LEP, Tevatron and LHC to examine the Higgs sector in our model and derive constraints on model parameters. We also discuss the implications of the model on the dark matter phenomenology., Comment: 18 pages, 7 figures

This paper considers a downlink satellite communication system where a satellite cluster, i.e., a satellite swarm consisting of one leader and multiple follower satellites, serves a ground terminal. The satellites in the cluster form either a linear or circular formation moving in a group and cooperatively send their signals by maximum ratio transmission precoding. We first conduct a coordinate transformation to effectively capture the relative positions of satellites in the cluster. Next, we derive an exact expression for the orbital configuration-dependent outage probability under the Nakagami fading by using the distribution of the sum of independent Gamma random variables. In addition, we obtain a simpler approximated expression for the outage probability with the help of second-order moment-matching. We also analyze asymptotic behavior in the high signal-to-noise ratio regime and the diversity order of the outage performance. Finally, we verify the analytical results through Monte Carlo simulations. Our analytical results provide the performance of satellite cluster-based communication systems based on specific orbital configurations, which can be used to design reliable satellite clusters in terms of cluster size, formation, and orbits., Comment: 5 pages, 5 figures, submitted to IEEE Transactions on Vehicular Technology

Because of its proximity and the large size of its black hole, M87 is one of the best targets for studying the launching mechanism of active galactic nucleus jets. Currently, magnetic fields are considered to be an essential factor in the launching and accelerating of the jet. However, current observational estimates of the magnetic field strength of the M87 jet are limited to the innermost part of the jet or to HST-1. No attempt has yet been made to measure the magnetic field strength in between. We aim to infer the magnetic field strength of the M87 jet out to a distance of several thousand $r_s$ by tracking the distance-dependent changes in the synchrotron spectrum of the jet from high-resolution very long baseline interferometry observations. In order to obtain high-quality spectral index maps, quasi-simultaneous observations at 22 and 43 GHz were conducted using the KVN and VERA Array (KaVA) and the VLBA. We compared the spectral index distributions obtained from the observations with a model and placed limits on the magnetic field strengths as a function of distance. The overall spectral morphology is broadly consistent over the course of these observations. The observed synchrotron spectrum rapidly steepens from $\alpha_{22-43 GHz}$ ~ -0.7 at ~ 2 mas to $\alpha_{22-43 GHz}$ ~ -2.5 at ~ 6 mas. A spectral index model in which nonthermal electron injections inside the jet decrease with distance can adequately reproduce the observed trend. This suggests the magnetic field strength of the jet at a distance of 2 - 10 mas (~ 900 $r_s$ - ~ 4500 $r_s$ in the deprojected distance) has a range of $B=(0.3 - 1.0 G)(z/2 mas)^{-0.73}$. Extrapolating to the EHT scale yields consistent results, suggesting that the majority of the magnetic flux of the jet near the black hole is preserved out to ~ 4500 $r_s$ without significant dissipation., Comment: 16 pages, 9 figures. Accepted for publication in Astronomy & Astrophysics

Recently, mega-constellations with a massive number of low Earth orbit (LEO) satellites are being considered as a possible solution for providing global coverage due to relatively low latency and high throughput compared to geosynchronous orbit satellites. However, as the number of satellites and operators participating in the LEO constellation increases, inter-satellite interference will become more severe, which may yield marginal improvement or even decrement in network throughput. In this article, we introduce the concept of satellite clusters that can enhance network performance through satellites' cooperative transmissions. The characteristics, formation types, and transmission schemes for the satellite clusters are highlighted. Simulation results evaluate the impact of clustering from coverage and capacity perspectives, showing that when the number of satellites is large, the performance of clustered networks outperforms the unclustered ones. The viable network architectures of the satellite cluster are proposed based on the 3GPP standard. Finally, the future applications of clustered satellite networks are discussed., Comment: 7 pages, 7 figures, 1 table, submitted to IEEE Vehicular Technology Magazine

We analyze the mass spectrum of the charged and neutral Higgs bosons in the framework of two Higgs doublet model (2HDM) in the light of the precision measurement of the $W$ boson mass by the CDF collaboration. We have considered the most general 2HDM potential with explicit CP violation in the Higgs basis which contains the three CP-mixed neutral mass eigenstates $H_1$, $H_2$, and $H_3$ with $M_{H_1}\leq M_{H_2}\leq M_{H_3}$. The high-precision CDF measurement of the $W$ boson mass is characterized by the large positive value of the $T$ parameter. By identifying the lightest neutral Higgs boson $H_1$ as the SM-like one discovered at the LHC, we find that it is necessary to have the mass splitting between the charged Higgs boson $H^\pm$ and the second heaviest neutral one $H_2$ to accommodate the sizable positive deviation of the $T$ parameter from its SM value of $0$. By combining the mass splitting between $H^\pm$ and $H_2$ with the theoretical constraints from the perturbative unitarity and for the Higgs potential to be bounded from below, we implement comprehensive analysis of the mass spectrum of the heavy Higgs bosons taking account of the effects of deviation from the alignment limit and also the mass splitting between $H_3$ and $H_2$. We further analyze the behavior of the heavy-Higgs mass spectrum according to the variation of the $T$ parameter. Finally, concentrating on the heavy mass region $M\equiv (M_{H_2}+M_{H^\pm})/2\gsim 500$ GeV in which we find a mass hierarchy of $M\gg M_{H_1}\sim |\Delta| \gsim \delta$ with $\Delta \equiv M_{H_2}-M_{H^\pm}$ and $\delta \equiv M_{H_3}-M_{H_2}$, we discuss some benchmarking scenarios for the searches of heavy Higgs bosons at future colliders such as the high luminosity option of the LHC and a 100 TeV hadron collider., Comment: 20pages, 7 figures, 1 table: version to appear in PRD

The East Asian VLBI Network (EAVN) is an international VLBI facility in East Asia and is operated under mutual collaboration between East Asian countries, as well as part of Southeast Asian and European countries. EAVN currently consists of 16 radio telescopes and three correlators located in China, Japan, and Korea, and is operated mainly at three frequency bands, 6.7, 22, and 43 GHz with the longest baseline length of 5078 km, resulting in the highest angular resolution of 0.28 milliarcseconds at 43 GHz. One of distinct capabilities of EAVN is multi-frequency simultaneous data reception at nine telescopes, which enable us to employ the frequency phase transfer technique to obtain better sensitivity at higher observing frequencies. EAVN started its open-use program in the second half of 2018, providing a total observing time of more than 1100 hours in a year. EAVN fills geographical gap in global VLBI array, resulting in enabling us to conduct contiguous high-resolution VLBI observations. EAVN has produced various scientific accomplishments especially in observations toward active galactic nuclei, evolved stars, and star-forming regions. These activities motivate us to initiate launch of the 'Global VLBI Alliance' to provide an opportunity of VLBI observation with the longest baselines on the earth., Comment: 27 pages, appeared in Galaxies special issue 'Challenges in Understanding Black Hole Powered Jets with VLBI' as an invited review

We aim to reveal the structure and kinematics of the Outer-Scutum-Centaurus (OSC) arm located on the far side of the Milky Way through very long baseline interferometry (VLBI) astrometry using KaVA, which is composed of KVN (Korean VLBI Network) and VERA (VLBI Exploration of Radio Astrometry). We report the proper motion of a 22 GHz H$_{2}$O maser source, which is associated with the star-forming region G034.84$-$00.95, to be ($\mu_{\alpha} \rm{cos}\delta$, $\mu_{\delta}$) = ($-$1.61$\pm$0.18, $-$4.29$\pm$0.16) mas yr$^{-1}$ in equatorial coordinates (J2000). We estimate the 2D kinematic distance to the source to be 18.6$\pm$1.0 kpc, which is derived from the variance-weighted average of kinematic distances with LSR velocity and the Galactic-longitude component of the measured proper motion. Our result places the source in the OSC arm and implies that G034.84$-$00.95 is moving away from the Galactic plane with a vertical velocity of $-$38$\pm$16 km s$^{-1}$. Since the H I supershell GS033+06$-$49 is located at a kinematic distance roughly equal to that of G034.84$-$00.95, it is expected that gas circulation occurs between the outer Galactic disk around G034.84$-$00.95 with a Galactocentric distance of 12.8$^{+1.0}_{-0.9}$ kpc and halo. We evaluate possible origins of the fast vertical motion of G034.84$-$00.95, which are (1) supernova explosions and (2) cloud collisions with the Galactic disk. However, neither of the possibilities are matched with the results of VLBI astrometry as well as spatial distributions of H II regions and H I gas., Comment: Accepted for publication in PASJ. 14 figures; 8 tables

This paper considers satellite eavesdroppers in uplink satellite communication systems where the eavesdroppers are randomly distributed at arbitrary altitudes according to homogeneous binomial point processes and attempt to overhear signals that a ground terminal transmits to a serving satellite. Non-colluding eavesdropping satellites are assumed, i.e., they do not cooperate with each other, so that their received signals are not combined but are decoded individually. Directional beamforming with two types of antennas: fixed- and steerable-beam antennas, is adopted at the eavesdropping satellites. The possible distribution cases for the eavesdropping satellites and the distributions of the distances between the terminal and the satellites are analyzed. The distributions of the signal-to-noise ratios (SNRs) at both the serving satellite and the most detrimental eavesdropping satellite are derived as closed-form expressions. The ergodic and outage secrecy capacities of the systems are derived with the secrecy outage probability using the SNR distributions. Simpler approximate expressions for the secrecy performance are obtained based on the Poisson limit theorem, and asymptotic analyses are also carried out in the high-SNR regime. Monte-Carlo simulations verify the analytical results for the secrecy performance. The analytical results are expected to be used to evaluate the secrecy performance and design secure satellite constellations by considering the impact of potential threats from malicious satellite eavesdroppers., Comment: 16 pages, 11 figures, 1 table, accepted by IEEE Transactions on Information Forensics and Security

We consider the implication of the recent CDF $W$-mass anomaly in the general framework of two Higgs doublet model. We find that the large deviation of the $S$ and $T$ parameters from their SM values of zero leads to the upper limit of about $1$ TeV on the heavy charged and neutral Higgs bosons when it is combined with the theoretical constraints from the perturbative unitarity and for the Higgs potential to be bounded from below., Comment: 12 pages, 2 figures, references added. To appear in PLB. Figures updated. arXiv admin note: substantial text overlap with arXiv:2110.03908

Sagittarius A* (Sgr A*), the Galactic Center supermassive black hole (SMBH), is one of the best targets to resolve the innermost region of SMBH with very long baseline interferometry (VLBI). In this study, we have carried out observations toward Sgr A* at 1.349 cm (22.223 GHz) and 6.950 mm (43.135 GHz) with the East Asian VLBI Network, as a part of the multi-wavelength campaign of the Event Horizon Telescope (EHT) in 2017 April. To mitigate scattering effects, the physically motivated scattering kernel model from Psaltis et al. (2018) and the scattering parameters from Johnson et al. (2018) have been applied. As a result, a single, symmetric Gaussian model well describes the intrinsic structure of Sgr A* at both wavelengths. From closure amplitudes, the major-axis sizes are ~704$\pm$102 $\mu$as (axial ratio $\sim$1.19$^{+0.24}_{-0.19}$) and $\sim$300$\pm$25 $\mu$as (axial ratio $\sim$1.28$\pm$0.2) at 1.349 cm and 6.95 mm respectively. Together with a quasi-simultaneous observation at 3.5 mm (86 GHz) by Issaoun et al. (2019), we show that the intrinsic size scales with observing wavelength as a power-law, with an index $\sim$1.2$\pm$0.2. Our results also provide estimates of the size and compact flux density at 1.3 mm, which can be incorporated into the analysis of the EHT observations. In terms of the origin of radio emission, we have compared the intrinsic structures with the accretion flow scenario, especially the radiatively inefficient accretion flow based on the Keplerian shell model. With this, we show that a nonthermal electron population is necessary to reproduce the source sizes., Comment: accepted to ApJ

We investigate the invisible decay of positronium to probe the fermionic light dark matter mediated by the dark $Z$ boson. Too tiny is the invisible decay rate of positronium through weak interaction in the standard model to be detected in the experiment. We show that it can be enhanced to be observed in the future if the dark matter is lighter than the electron in the dark $Z$ model. We also compute the relic abundance of such light dark matter and discuss the Big Bang Nucleosynthesis constraint with an alternative thermal history scenario., Comment: 8 pages, 4 figures

In this paper, we consider downlink low Earth orbit (LEO) satellite communication systems where multiple LEO satellites are uniformly distributed over a sphere at a certain altitude according to a homogeneous binomial point process (BPP). Based on the characteristics of the BPP, we analyze the distance distributions and the distribution cases for the serving satellite. We analytically derive the exact outage probability, and its approximated expression is obtained using the Poisson limit theorem. With these derived expressions, the system throughput maximization problem is formulated under the satellite-visibility and outage constraints. To solve this problem, we reformulate it with bounded feasible sets and propose an iterative algorithm to obtain near-optimal solutions. Simulation results perfectly match the derived exact expressions for the outage probability and system throughput. The analytical results of the approximated expressions are fairly close to those of the exact ones. It is also shown that the proposed algorithm for the throughput maximization is very close to the optimal performance obtained by a two-dimensional exhaustive search., Comment: 15 pages, 14 figures, 1 table, accepted by IEEE Transactions on Communications

The East Asian very-long-baseline interferometry (VLBI) Network (EAVN) is a rapidly evolving international VLBI array that is currently promoted under joint efforts among China, Japan, and Korea. EAVN aims at forming a joint VLBI Network by combining a large number of radio telescopes distributed over East Asian regions. After the combination of the Korean VLBI Network (KVN) and the VLBI Exploration of Radio Astrometry (VERA) into KaVA, further expansion with the joint array in East Asia is actively promoted. Here we report the first imaging results (at 22 and 43 GHz) of bright radio sources obtained with KaVA connected to Tianma 65-m and Nanshan 26-m Radio Telescopes in China. To test the EAVN imaging performance for different sources, we observed four active galactic nuclei (AGN) having different brightness and morphology. As a result, we confirmed that Tianma 65-m Radio Telescope (TMRT) significantly enhances the overall array sensitivity, a factor of 4 improvement in baseline sensitivity and 2 in image dynamic range compared to the case of KaVA only. The addition of Nanshan 26-m Radio Telescope (NSRT) further doubled the east-west angular resolution. With the resulting high-dynamic-range, high-resolution images with EAVN (KaVA+TMRT+NSRT), various fine-scale structures in our targets, such as the counter-jet in M87, a kink-like morphology of the 3C273 jet and the weak emission in other sources, are successfully detected. This demonstrates the powerful capability of EAVN to study AGN jets and to achieve other science goals in general. Ongoing expansion of EAVN will further enhance the angular resolution, detection sensitivity and frequency coverage of the network., Comment: 19 pages, 9 figures, accepted by Research in Astronomy and Astrophysics (RAA)

We derive a general formula for the inertia tensor of a three-body system. By employing three independent Lagrange undetermined multipliers to express the vectors corresponding to the sides in terms of the position vectors of the vertices, we present the general covariant expression for the inertia tensor of the three particles of different masses. If $m_a/a=m_b/b=m_c/c=\rho$, then the center of mass coincides with the incenter of the triangle and the moment of inertia about the normal axis passing the center of mass is $I=\rho abc$, where $m_a$, $m_b$, and $m_c$ are the masses of the particles at $A$, $B$, and $C$, respectively, and $a$, $b$, and $c$ are the lengths of the line segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively. The derivation and the corresponding results are closely related to the famous Heron's formula for the area of a triangle., Comment: 14 pages, 3 figures