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At least 217 records · Page 12

Searching for Interstellar Quantum Communications

The modern search for extraterrestrial intelligence began with the seminal publications of Cocconi & Morrison and Schwartz & Townes, who proposed searching for narrowband signals in the radio spectrum and optical laser pulses. Over the last six decades, more than 100 dedicated search programs have targeted these wavelengths, all with null results. All of these campaigns searched for classical communications, that is, for a significant number of photons above a noise threshold, with the assumption of a pattern encoded in time and/or frequency space. I argue that future searches should also target quantum communications. They are preferred over classical communications with regard to security and information efficiency, and they would have escaped detection in all previous searches. The measurement of Fock state photons or squeezed light would indicate the artificiality of a signal. I show that quantum coherence is feasible over interstellar distances and explain for the first time how astronomers can search for quantum transmissions sent by ETI to Earth using commercially available telescopes and receiver equipment.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

OGLE-2018-BLG-0567Lb and OGLE-2018-BLG-0962Lb: Two Microlensing Planets through the Planetary-caustic Channel

We present the analyses of two microlensing events, OGLE-2018-BLG-0567 and OGLE-2018-BLG-0962. In both events, the short-lasting anomalies were densely and continuously covered by two high-cadence surveys. The light-curve modeling indicates that the anomalies are generated by source crossings over the planetary caustics induced by planetary companions to the hosts. The estimated planet/host separation (scaled to the angular Einstein radius θ {sub E}) and mass ratio are (s, q × 10{sup 3}) = (1.81 ± 0.02, 1.24 ± 0.07) and (s, q × 10{sup 3}) = (1.25 ± 0.03, 2.38 ± 0.08), respectively. From Bayesian analyses, we estimate the host and planet masses as (M{sub h},M{sub p})=(0.25{sub −0.13}{sup +0.27} M{sub ⊙},0.32{sub −0.17}{sup +0.34} M{sub J}) and (M{sub h},M{sub p})=(0.54{sub −0.28}{sup +0.33} M{sub ⊙},1.34{sub −0.70}{sup +0.82} M{sub J}), respectively. These planetary systems are located at a distance of 7.06{sub −1.15}{sup +0.93} kpc for OGLE-2018-BLG-0567 and 6.50{sub −1.75}{sup +1.06} kpc for OGLE-2018-BLG-0962, suggesting that they are likely to be near the Galactic bulge. The two events prove the capability of current high-cadence surveys for finding planets through the planetary-caustic channel. We find that most published planetary-caustic planets are found in Hollywood events in which the source size strongly contributes to the anomaly cross-section relative to the size of the caustic.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

KMT-2019-BLG-0371 and the Limits of Bayesian Analysis

We show that the perturbation at the peak of the light curve of microlensing event KMT-2019-BLG-0371 is explained by a model with a mass ratio between the host star and planet of q ∼ 0.08. Due to the short event duration (t {sub E} ∼ 6.5 days), the secondary object in this system could potentially be a massive giant planet. A Bayesian analysis shows that the system most likely consists of a host star with a mass M{sub h}=0.09{sub −0.05}{sup +0.14}M{sub ⊙} and a massive giant planet with a mass M{sub p}=7.70{sub −3.90}{sup +11.34}M{sub Jup}. However, the interpretation of the secondary as a planet (i.e., as having M {sub p} < 13M {sub Jup}) rests entirely on the Bayesian analysis. Motivated by this event, we conduct an investigation to determine which constraints meaningfully affect Bayesian analyses for microlensing events. We find that the masses inferred from such a Bayesian analysis are determined almost entirely by the measured value of θ {sub E} and are relatively insensitive to other factors such as the direction of the event (ℓ, b), the lens–source relative proper motion μ {sub rel}, or the specific Galactic model prior.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Limiting Behavior of Asteroid Obliquity and Spin Using a Semi-analytic Thermal Model of the YORP Effect

The Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect governs the spin evolution of small asteroids. The axial component of YORP, which alters the rotation rate of the asteroid, is mostly independent of its thermal inertia, while the obliquity component is very sensitive to the thermal model of the asteroid. Here, we develop a semi-analytic theory for the obliquity component of YORP. We integrate an approximate thermal model over the surface of an asteroid, and find an analytic expression for the obliquity component in terms of two YORP coefficients. This approach allows us to investigate the overall evolution of asteroid rotation state, and to generalize the results previously obtained in the case of zero thermal inertia. The proposed theory also explains how a nonzero obliquity component of YORP originates even for a symmetric asteroid, due to its finite thermal inertia. In many cases, this causes equatorial planes of asteroids to align with their orbital planes. The studied nontrivial behavior of YORP as a function of thermal model allows for a new kind of rotational equilibria, which can have important evolutionary consequences for asteroids.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

A Canonical Transformation to Eliminate Resonant Perturbations. I

We study dynamical systems that admit action-angle variables at leading order, which are subject to nearly resonant perturbations. If the frequencies characterizing the unperturbed system are not in resonance, the long-term dynamical evolution may be integrated by orbit-averaging over the high-frequency angles, thereby evolving the orbit-averaged effect of the perturbations. It is well known that such integrators may be constructed via a canonical transformation, which eliminates the high-frequency variables from the orbit-averaged quantities. An example of this algorithm in celestial mechanics is the von Zeipel transformation. However, if the perturbations are inside or close to a resonance, i.e., the frequencies of the unperturbed system are commensurate; these canonical transformations are subject to divergences. We introduce a canonical transformation that eliminates the high-frequency phase variables in the Hamiltonian without encountering divergences. This leads to a well-behaved symplectic integrator. We demonstrate the algorithm through two examples: a resonantly perturbed harmonic oscillator and the gravitational three-body problem in mean motion resonance.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

OGLE-2017-BLG-1434Lb: Confirmation of a Cold Super-Earth Using Keck Adaptive Optics

The microlensing event OGLE-2017-BLG-1434 features a cold super-Earth planet that is 1 of 11 microlensing planets with a planet–host-star mass ratio of q < 1 × 10{sup -4}. We provide an additional mass–distance constraint on the lens host using near-infrared adaptive optics photometry from Keck/NIRC2. We are able to determine a flux excess of K {sub L} = 16.96 ± 0.11, which most likely comes entirely from the lens star. Combining this with constraints from the large Einstein ring radius, θ {sub E} = 1.40 ± 0.09 mas, and OGLE parallax we confirm this event as a super-Earth with a mass of m {sub p} = 4.43 ± 0.25M {sub ⊕}. This system lies at a distance of D {sub L} = 0.86 ± 0.05 kpc from Earth and the lens star has a mass of M {sub L} = 0.234 ± 0.012M {sub ⊙}. We confirm that with a star–planet mass ratio of q = 0.57 × 10{sup -4}, OGLE-2017-BLG-1434 lies near the inflexion point of the planet–host mass-ratio power law.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Mapping Stellar Surfaces. II. An Interpretable Gaussian Process Model for Light Curves

The use of Gaussian processes (GPs) as models for astronomical time series data sets has recently become almost ubiquitous, given their ease of use and flexibility. In particular, GPs excel at marginalization over the stellar signal when the variability due to starspots is treated as a nuisance, as in exoplanet transit modeling. However, these effective models are less useful in cases where the starspot signal is of primary interest, since it is not obvious how the parameters of the GP relate to physical parameters like the spot size, contrast, and latitudinal distribution. Instead, it is common practice to explicitly model the effect of individual starspots on the light curve and attempt to infer their properties via optimization or posterior inference. Unfortunately, this process is ill-posed and often computationally intractable when applied to stars with more than a few spots and/or to ensembles of many stars. Here we derive a closed-form expression for a GP that describes the light curve of a rotating, evolving stellar surface conditioned on a given distribution of starspot sizes, contrasts, and latitudes. We demonstrate that this model is correctly calibrated, allowing one to robustly infer physical parameters of interest from one or more light curves, including the typical spot radii and latitudes. Our GP has far-ranging implications for understanding the variability and magnetic activity of stars from light curves and radial velocity measurements, as well as for modeling correlated noise in exoplanet searches. Our implementation is efficient, user-friendly, and open-source, available in the package starry-process.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Meteoroid Stream of Comet C/1961 T1 (Seki) and Its Relation to the December ρ-Virginids and γ-Sagittariids

We answer the question of whether long-period comet C/1961 T1 (Seki) is the parent body of a meteor shower observed in Earth’s atmosphere. We model discrete parts of a meteoroid stream associated with this comet and follow the orbital evolution of its particles via a numerical integration of their orbits. The gravitational perturbations of major planets and the Poynting–Robertson effect, which influences the motion of the particles, are taken into account. We predict two showers which are related to C/1961 T1. One of these showers was identified with the December ρ-Virginids, #502. This identification confirms an earlier suggested relationship between the shower and the comet. The second predicted shower was identified with the γ-Sagittariids, #657. The radiant areas of both showers are symmetrical in respect to the apex of Earth’s heliocentric motion.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Rotational Velocities of Am and Non-chemical-peculiar Stars Based on Kepler and LAMOST DR5

Rotation is a critical physical process operating in the formation of Am stars. There is a strong correlation between low-velocity rotation and chemical peculiarity. However, the existence of many non-chemical-peculiar slow rotators challenges the understanding of Am stars. The purpose of our work is to search for low-velocity rotating non-chemical-peculiar A-type stars and Am stars and to make a comparative analysis. In this paper, we pick out a sample from the LAMOST–Kepler project, including 21 Am stars, 125 non-chemical-peculiar slow rotators, and 53 non-chemical-peculiar fast rotators. We calculate the rotational frequencies through a periodic change of light curves caused by inhomogeneous stellar surfaces and then obtain the rotational velocities. For slow rotators, the age of Am stars is statistically younger than that of non-chemical-peculiar stars in the same temperature bin. In the comparison of the period, the average amplitude, and stellar mass of Am and non-chemical-peculiar stars, we discover that there is no difference in the photometric variability between Am and non-chemical-peculiar stars, which implies similar inhomogeneities on the surfaces. The average amplitude of non-chemical-peculiar stars has a downward trend with the increase of effective temperature and stellar mass, which is consistent with the theoretical prediction caused by weak dynamo-generated magnetic fields in A-type stars. In addition, we confirm four non-chemical-peculiar stars that have flares by checking field-of-view images, pixel images, and pixel-level light curves.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Photometric and Kinematic Study of the Open Clusters SAI 44 and SAI 45

We carry out a detailed photometric and kinematic study of the poorly studied sparse open clusters SAI 44 and SAI 45 using ground-based BVR {sub c} I {sub c} data supplemented by archival data from Gaia eDR3 and Pan-STARRS. The stellar memberships are determined using a statistical method based on Gaia eDR3 kinematic data, and we found 204 members in SAI 44 while only 74 members are identified in SAI 45. The average distances to SAI 44 and SAI 45 are calculated to be 3670 ± 184 and 1668 ± 47 pc. The logarithmic age of the clusters are determined to be 8.82 ± 0.10 and 9.07 ± 0.10 yr for SAI 44 and SAI 45, respectively. The color–magnitude diagram of SAI 45 hosts an extended main-sequence turnoff (eMSTO). The apparent age spread is found to be similar to the apparent age spread predicted on the basis of the age spread and cluster age relation predicted by rotation models. This indicates that eMSTO is a stellar evolution rather than star formation phenomenon in SAI 45. We conclude that eMSTO in SAI 45 is mainly caused by the different rotation rates of stars as the SYCLIST synthetic population with different rotation rates was able to reproduce the observed eMSTO, and stars in the red part of the eMSTO were preferentially concentrated in the inner region, which again hints at different rotations being the reason for the extension in the upper MS. This finding supports the theory attributing the origin of eMSTO to the different rotations of eMSTO stars. The mass function slopes are obtained as -2.24 ± 0.66 and -2.58 ± 3.20 in the mass rages 2.426–0.990 M{sub ⊙} and 2.167–1.202 M{sub ⊙} for SAI 44 and SAI 45, respectively. SAI 44 exhibits the signature of mass segregation while we found weak evidence of mass segregation in SAI 45 possibly due to tidal stripping. The dynamical relaxation times of these clusters indicate that both clusters are in a dynamically relaxed state. Using the AD-diagram method, the apex coordinates are found to be ([[Formula]], -[[Formula]]) for SAI 44 and (-[[Formula]], -[[Formula]]) for SAI 45. The average space velocity components of the clusters SAI 44 and SAI 45 are calculated in units of km s{sup -1} as (-15.14 ± 3.90, -19.43 ± 4.41, -20.85 ± 4.57) and (28.13 ± 5.30, -9.78 ± 3.13, -19.59 ± 4.43), respectively.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Thermophysical Modeling of 20 Themis Family Asteroids with WISE/NEOWISE Observations

Themis family is one of the largest and oldest asteroid populations in the main belt. Water ice may widely exist on the parent body (24) Themis. In this work, we employ the Advanced Thermophysical Model as well as midinfrared measurements from NASA’s Wide-field Infrared Survey Explorer to explore thermal parameters of 20 Themis family members. Here we show that the average thermal inertia and geometric albedo are 39.5 ± 26.0 J m{sup −2} s{sup −1/2} K{sup −1} and 0.067 ± 0.018, respectively. The family members have a relatively moderate roughness fraction on their surfaces. We find that the relatively low albedos of Themis members are consistent with the typical values of B-type and C-type asteroids. As mentioned, the Themis family bears a very low thermal inertia, which indicates a fine and mature regolith on their surfaces. The resemblance of thermal inertia and geometric albedo of Themis members may reveal their close connection in origin and evolution. In addition, we present the compared results of thermal parameters for several prominent families.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Understanding the Angular Momentum Evolution of T Tauri and Herbig Ae/Be Stars

We investigate a sample of six Herbig Ae/Be stars belonging to the Orion OB1 association, as well as 73 low-mass objects, members of the σ Orionis cluster, in order to explore the angular momentum evolution at early stages of evolution, and its possible connection with main-sequence Ap/Bp magnetic stars. Using FIES and HECTOCHELLE spectra, we obtain projected rotational velocities through two independent methods. Individual masses, radii, and ages are computed using evolutionary models, distance, and cluster extinction. Under the assumption that similar physical processes operate in both T Tauri and Herbig Ae/Be stars, we construct snapshots of the protostar’s rotation against mass during the first 10 Myr with the aid of a rotational model that includes a variable disk lifetime, changes in the stellar moment of inertia, a dipolar magnetic field with variable strength, and angular momentum loss through stellar winds powered by accretion. We use these snapshots, as well as the rotational data, to infer a plausible scenario for the angular momentum evolution. We find that magnetic field strengths of a few kilo-Gauss at 3 Myr are required to match the rotational velocities of both groups of stars. Models with masses between 2 M {sub ⊙} and 3 M {sub ⊙} display larger angular momentum values by a factor of ~3, in comparison to stars of similar spectral types on the main sequence. Even though some quantitative estimates on this dramatic decrease with age for Ap/Bp magnetic main-sequence stars are presented, the results obtained for the angular momentum evolution do not explain their low rotation rates.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Long-period Jovian Tilts the Orbits of Two sub-Neptunes Relative to Stellar Spin Axis in Kepler-129

We present the discovery of Kepler-129 d (P{sub d}=7.2{sub -0.3}{sup +0.4} yr, msini{sub d}=8.3{sub -0.7}{sup +1.1} M{sub Jup}, e{sub d}=0.15{sub -0.05}{sup +0.07}) based on six years of radial-velocity observations from Keck/HIRES. Kepler-129 also hosts two transiting sub-Neptunes: Kepler-129 b (P {sub b} = 15.79 days, r {sub b} = 2.40 ± 0.04 R {sub ⊕}) and Kepler-129 c (P {sub c} = 82.20 days, r {sub c} = 2.52 ± 0.07 R {sub ⊕}) for which we measure masses of m {sub b} < 20 M {sub ⊕} and m{sub c}=43{sub -12}{sup +13} M{sub ⊕}. Kepler-129 is a hierarchical system consisting of two tightly packed inner planets and a massive external companion. In such a system, two inner planets precess around the orbital normal of the outer companion, causing their inclinations to oscillate with time. Based on an asteroseismic analysis of Kepler data, we find tentative evidence that Kepler-129 b and c are misaligned with stellar spin axis by ≳38°, which could be torqued by Kepler-129 d if it is inclined by ≳19° relative to inner planets. Using N-body simulations, we provide additional constraints on the mutual inclination between Kepler-129 d and inner planets by estimating the fraction of time during which two inner planets both transit. The probability that two planets both transit decreases as their misalignment with Kepler-129 d increases. We also find a more massive Kepler-129 c enables the two inner planets to become strongly coupled and more resistant to perturbations from Kepler-129 d. The unusually high mass of Kepler-129 c provides a valuable benchmark for both planetary dynamics and interior structure, since the best-fit mass is consistent with this 2.5 R {sub ⊕} planet having a rocky surface.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NGC 6845 Revisited: Stellar Kinematics on the Early Type Member Galaxies

We observed the compact group of galaxies NGC 6845 with GMOS-S imaging and long-slit spectroscopy. The group itself is comprised of two spiral galaxies and two S0 galaxies, and shows signs of gravitational interaction among them. While the spiral galaxies have been already well studied, the two S0 galaxies have received little attention. We investigate the morphology and stellar kinematics of the two S0 galaxies in detail in order to derive the value for their mass and subsequently incorporate them into the mass model for the group. It was found that the internal kinematics of the largest S0 (NGC 6845 C) is mostly regular, while the smallest S0 (NGC 6845 D) shows some hint of interaction, most likely with NGC 6845 C and NGC 6845 A. The new determination of the dynamical mass of the group implies that the group mass has to be 10 times the combined mass of the individual members according to their internal kinematics. The results point to an early stage of the group interaction and also to the possibility that it has still not reached virial equilibrium. From the galaxy evolution viewpoint, this group represents an example of the case of the transformation of Spirals into S0 galaxies in dense environments.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Visible-light Phase Curves from the Second Year of the TESS Primary Mission

We carried out a systematic study of full-orbit phase curves for known transiting systems in the northern ecliptic sky that were observed during Year 2 of the TESS primary mission. We applied the same methodology for target selection, data processing, and light-curve fitting as we did in our Year 1 study. Out of the 15 transiting systems selected for analysis, seven—HAT-P-7, KELT-1, KELT-9, KELT-16, KELT-20, Kepler-13A, and WASP-12—show statistically significant secondary eclipses and day–night atmospheric brightness modulations. Small eastward dayside hot-spot offsets were measured for KELT-9b and WASP-12b. KELT-1, Kepler-13A, and WASP-12 show additional phase-curve variability attributed to the tidal distortion of the host star; the amplitudes of these signals are consistent with theoretical predictions. We combined occultation measurements from TESS and Spitzer to compute dayside brightness temperatures, TESS-band geometric albedos, Bond albedos, and phase integrals for several systems. The new albedo values solidify the previously reported trend between dayside temperature and geometric albedo for planets with 1500 K < T {sub day} < 3000 K. For Kepler-13Ab, we carried out an atmospheric retrieval of the full secondary eclipse spectrum, which revealed a noninverted temperature–pressure profile, significant H{sub 2}O and K absorption in the near-infrared, evidence for strong optical atmospheric opacity due to sodium, and a confirmation of the high geometric albedo inferred from our simpler analysis. We explore the implications of the phase integrals (ratios of Bond to geometric albedos) for understanding exoplanet clouds. We also report updated transit ephemerides for all of the systems studied in this work.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

VLBI Data Processing on Coronal Radio-sounding Experiments of Mars Express

The ESA’s Mars Express solar corona experiments were performed at two solar conjunctions in the years 2015 and 2017 by a number of radio telescopes in the European VLBI Network. This paper presents the methods to measure the frequency and phase fluctuations of the spacecraft radio signal, and the applications to study the characteristics of the plasma turbulence effects on the signal at a single station and at multiple stations via cross correlation. The power spectra of the frequency fluctuations observed between 4.9 and 76.3 R {sub s} have a power-law shape close to a Kolmogorov spectrum over the frequency interval ν {sub lo} < ν < ν {sub up}, where the nominal value of ν {sub lo} is set to 3 mHz and ν {sub up} is in the range of 0.03–0.15 Hz. The rms of the frequency fluctuations is presented as a function of the heliocentric distance. Furthermore, we analyze the variations of the electron column density fluctuations at solar offsets 4.9 R {sub s} and 9.9 R {sub s} and the cross-correlation products between the VLBI stations. The power density of the differential fluctuations between different stations decreases at ν < 0.01 Hz. Finally, the fast flow speeds of solar wind >700 km s{sup -1} are derived from the cross correlation of frequency fluctuations at ν < 0.01 Hz. The fast flow speeds of solar wind correspond to the high heliolatitude of the coronal region that the radio rays passed. The VLBI observations and analysis methods can be used to study the electron column density fluctuations and the turbulence at multiple spatial points in the inner solar wind by providing multiple lines of sight between the Earth and the spacecraft.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Atmospheric Transmission Spectroscopy of Hot Jupiter KELT-10b using Synthetic Telluric Correction Software

High-resolution spectroscopic visible data were obtained with the Ultraviolet and Visible Echelle Spectrograph on the Very Large Telescope. Our goal was to analyze the data in an effort to detect the presence of sodium in the atmosphere of hot Jupiter exoplanet KELT-10b, as well as characterize the orbit of the planet via the Rossiter-McLaughlin effect. Eighty spectra were collected during a single transit of KELT-10b. After standard spectroscopic calibration using ESO-Reflex, the synthetic telluric modeling software molecfit was applied to remove terrestrial atmospheric effects, and to refine the wavelength calibration. Sodium is recognized by its characteristic absorption doublet located at 5895.924 and 5889.951 Å, which can be seen in the planet atmosphere transmission spectrum and through excess absorption during the transit. The radial velocity of the host star was analyzed by measuring the average shift of absorption features from spectrum to spectrum. Our results indicate a sodium detection in the planet transmission spectrum with a line contrast of 0.66% and 0.43% ± 0.09% for the sodium DII and DI lines, respectively. Excess absorption measurements agree to within one half combined standard deviation between the planet transmission spectrum (0.143% ± 0.020%, a 7σ detection) and during the time series (0.124% ± 0.034%, a 3.6σ detection) in a band 1.25 Å wide. The wavelength grid corrections provided by molecfit were insufficient to determine radial velocities and measure the Rossiter-McLaughlin effect.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Sanford Underground Research Facility’s approach to school education, community activities, and public outreach

The Sanford Underground Research Facility (SURF) is the deepest underground science facility in the United States. SURF hosts world-leading experiments in neutrino, astroparticle and nuclear physics, as well as projects in biology, geology, and engineering, and is home to a major excavation project making space for Fermi National Accelerator Laboratory’s Long-Baseline Neutrino Facility (LBNF), which will power the Deep Underground Neutrino Experiment (DUNE). An emphasis on outreach and education is embedded in SURF’s mission statement: “to advance world-class science and inspire learning across generations.” To achieve this mission, SURF goes beyond established science communication methods, including operating an open-to-the-public visitor center, hosting multiple public outreach events per month, and an annual city-wide science festival. Furthermore, SURF is training K-12 science educators, developing school curriculum units, and providing classroom materials, based on science researched at the laboratory. The strategic approach, specific methods, and successful outcomes of these programs, which are based on SURF’s science, location, and community, may serve as examples for effective science education, public outreach, and community engagement.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗