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At least 415 records · Page 23

Apparatus for providing a servo drive signal in a high-speed stepping interferometer

An analog voltage approximately linearly proportional to a desired offset from the present null position of a moving mirror in an interferometer is applied to the mirror moving means. As the mirror moves to the next null position, as determined by the analog voltage, the fringes of a laser reference interference pattern are detected. At the occurrence of each fringe the analog voltage is reduced proportionally so that when the next null position is reached, this driving analog is effectively zero. A binary up/down counter, by its internal count, causes a digital/analog converter to supply the analog voltage to the mirror moving means. Fringe detection and direction of movement logic cause the binary up/down counter to be decremented from its offset count as the mirror is moved to the new null position. Undesirable movement of the mirror due to vibration or other sources causes a correcting drive signal to be applied to the mirror moving means that is proportional to the distance of movement.

Schindler, R. A.↗

The HST Lightcurve of (486958) 2014 MU69

We report Hubble Space Telescope (HST) lightcurve observations of the New Horizons spacecraft encounter Kuiper Belt object (KBO) (486958) 2014 MU69 acquired near opposition in July 2017. In order to plan the optimum flyby sequence the New Horizons mission planners needed to learn as much as possible about the target in advance of the encounter. Specifically, from lightcurve data, encounter timing could be adjusted to accommodate a highly elongated, binary, or rapidly rotating target. HST astrometric (Porter et al., 2018) and stellar occultation (Buie et al., 2018) observations constrained MU69's orbit and diameter (21–41 km for an albedo of 0.15–0.04), respectively. Photometry from the astrometric dataset suggested a variability of ≥0.3 mags, but they did not determine the period or provide shape information. To that end we strategically spaced 24 HST orbits over 9 days to investigate rotation periods from approximately 2–100 h and to better constrain the lightcurve amplitude. Until New Horizons detected MU69 in its optical navigation images beginning in August 2018, this HST lightcurve campaign provided the most accurate photometry to date. The mean variation in our data is 0.15 magnitudes which suggests that MU69 is either nearly spherical (a:b axis ratio of 1:1.15), or its pole vector is pointed near the line of sight to Earth; this interpretation does not preclude a near-contact binary or bi-lobed object. However, image stacks do conclude that MU69 does not have a binary companion ≥2000 km with a sensitivity to 29th magnitude (an object a few km in size for an albedo of 0.04–0.15). Our data are not of sufficient signal to noise to uniquely determine the period or amplitude, however, they did provide the necessary information for spacecraft planning. We report with confidence that MU69 is not both rapidly rotating and highly elongated (which we define as a lightcurve amplitude ≥ 0.5 magnitude). Since this paper is being published post fly-by, we note that our results are consistent with the fly-by imagery and orientation of MU69 (Stern et al., 2019). The combined dataset also suggests that within the KBO lightcurve literature there are likely other objects which share a geometric configuration like MU69 resulting in an underestimate of the contact binary fraction for the Cold Classical Kuiper Belt.

S D Benecchi↗

Forward Modeling the Orbits of Companions to Pulsating Stars from Their Light Travel Time Variations

Mutual gravitation between a pulsating star and an orbital companion leads to a time-dependent variation in path length for starlight traveling to Earth. These variations can be used for coherently pulsating stars, such as the δ Scuti variables, to constrain the masses and orbits of their companions. Observing these variations for δ Scuti stars has previously relied on subdividing the light curve and measuring the average pulsation phase in equally sized subdivisions, which leads to undersampling near periapsis. We introduce a new approach that simultaneously forward models each sample in the light curve and show that this method improves upon current sensitivity limits—especially in the case of highly eccentric and short-period binaries. We find that this approach is sensitive enough to observe Jupiter mass planets around δ Scuti stars under ideal conditions, and use gravity-mode pulsations in the subdwarf B star KIC 7668647 to detect its companion without radial velocity data. We further provide robust detection limits as a function of the signal-to-noise ratio of the pulsation mode and determine that the minimum detectable light travel time amplitude for a typical Kepler δ Scuti is around 2 s. This new method significantly enhances the application of light travel time variations to detecting short-period binaries with pulsating components, and pulsating A-type exoplanet host stars, especially as a tool for eliminating false positives.

79 ASTRONOMY AND ASTROPHYSICS↗

Digital parallel-to-series pulse-train converter

Circuit converts number represented as two level signal on n-bit lines to series of pulses on one of two lines, depending on sign of number. Converter accepts parallel binary input data and produces number of output pulses equal to number represented by input data.

Hussey, J.↗

Punctured Parallel and Serial Concatenated Convolutional Codes for BPSK/QPSK Channels

As available bandwidth for communication applications becomes scarce, bandwidth-efficient modulation and coding schemes become ever important. Since their discovery in 1993, turbo codes (parallel concatenated convolutional codes) have been the center of the attention in the coding community because of their bit error rate performance near the Shannon limit. Serial concatenated convolutional codes have also been shown to be as powerful as turbo codes. In this dissertation, we introduce algorithms for designing bandwidth-efficient rate r = k/(k + 1),k = 2, 3,..., 16, parallel and rate 3/4, 7/8, and 15/16 serial concatenated convolutional codes via puncturing for BPSK/QPSK (Binary Phase Shift Keying/Quadrature Phase Shift Keying) channels. Both parallel and serial concatenated convolutional codes have initially, steep bit error rate versus signal-to-noise ratio slope (called the -"cliff region"). However, this steep slope changes to a moderate slope with increasing signal-to-noise ratio, where the slope is characterized by the weight spectrum of the code. The region after the cliff region is called the "error rate floor" which dominates the behavior of these codes in moderate to high signal-to-noise ratios. Our goal is to design high rate parallel and serial concatenated convolutional codes while minimizing the error rate floor effect. The design algorithm includes an interleaver enhancement procedure and finds the polynomial sets (only for parallel concatenated convolutional codes) and the puncturing schemes that achieve the lowest bit error rate performance around the floor for the code rates of interest.

Acikel, Omer Fatih↗

Implications for the Origin of GRB 051103 from LIGO Observations

We present the results of a LIGO search for gravitational waves (GWs) associated with GRB 051103, a short-duration hard-spectrum gamma-ray burst whose electromagnetically determined sky position is coincident with the spiral galaxy M81, which is 3.6Mpc from Earth. Possible progenitors for short-hard GRBs include compact object mergers and soft gamma repeater (SGR) giant flares. A merger progenitor would produce a characteristic GW signal that should be detectable at the distance of M81, while GW emission from an SGR is not expected to be detectable at that distance. We found no evidence of a GW signal associated with GRB 051103. Assuming weakly beamed gamma-ray emission with a jet semi-angle of 30. we exclude a binary neutron star merger in M81 as the progenitor with a confidence of 98%. Neutron star-black hole mergers are excluded with > 99% confidence. If the event occurred in M81 our findings support the hypothesis that GRB 051103 was due to an SGR giant flare, making it the most distant extragalactic magnetar observed to date.

Bizouard, M. A.↗

The color and binarity of (486958) 2014 MU69 and other long-range New Horizons Kuiper Belt targets

The Hubble Space Telescope (HST) measured the colors of eight Kuiper Belt Objects (KBOs) that will be observed by the New Horizons spacecraft including its 2019 close fly-by target the Cold Classical KBO (486958) 2014 MU(69). We find that the photometric colors of all eight objects are red, typical of the Cold Classical dynamical population within which most reside. Because 2014 MU(69) has a similar color to that of other KBOs in the Cold Classical region of the Kuiper Belt, it may be possible to use the upcoming high-resolution New Horizons observations of 2014 MU(69) to draw conclusions about the greater Cold Classical population. Additionally, HST found none of these KBOs to be binary within separations of ~0.06 arcsec (~2000 km at 44 AU range) and Δm ≤ 0.5. This conclusion is consistent with the lower fraction of binaries found at relatively wide separations. A few objects appear to have significant photometric variability, but our observations are not of sufficient signal-to-noise or time duration for further interpretation.

S.D. Benecchi↗

Subphotospheric Emission from Short Gamma-Ray Bursts: Protons Mold the Multimessenger Signals

Abstract The origin of the observed Band-like photon spectrum in short gamma-ray bursts (sGRBs) is a long-standing mystery. We carry out the first general relativistic magnetohydrodynamic simulation of an sGRB jet with initial magnetizationσ 0 = 150 in dynamical ejecta from a binary merger. From this simulation, we identify regions along the jet of efficient energy dissipation due to magnetic reconnection and collisionless subshocks. Taking into account electron and proton acceleration processes, we solve for the first time the coupled transport equations for photons, electrons, protons, neutrinos, and intermediate particle species up to close to the photosphere (i.e., up to 1 × 10 12 cm), accounting for all relevant radiative and cooling processes. We find that the subphotospheric multimessenger signals carry strong signatures of the hadronic interactions and their resulting particle cascades. Importantly, the spectral energy distribution of photons is significantly distorted with respect to the Wien one, commonly assumed below the photosphere. Our findings suggest that the bulk of the nonthermal photon spectrum observed in sGRBs can stem from hadronic processes occurring below the photosphere and previously neglected, with an accompanying energy flux of neutrinos peaking in the GeV energy range.

Astronomy & Astrophysics↗

Impact of O4 future detections on the determination of the dense matter equations of state

Here, in view of the next LIGO-Virgo-KAGRA Observing period O4 (to start in Spring 2023), we address the question of the ability of the interferometers network to discriminate among different neutron stars equation of states better than what was possible with the observation of the binary neutron stars merger GW170817. We show that the observation of an event similar to GW170817 during O4 would allow us to resolve the dimensionless effective tidal deformability $\tilde{Λ}$ within an uncertainty 7 times better than the one obtained in O2. Thanks to the expected increase in sensitivities, we show that any GW170817-like single-event within a distance of 100 Mpc would imply significantly improved constraints of the neutron stars equations of state. We also illustrate the important impact of the noise in the analysis of the signal, showing how it can impact the effective tidal deformability probability density function for large signal-to-noise ratio.

79 ASTRONOMY AND ASTROPHYSICS↗

GROWTH on S190814bv: Deep Synoptic Limits on the Optical/Near-infrared Counterpart to a Neutron Star–Black Hole Merger

On 2019 August 14, the Advanced LIGO and Virgo interferometers detected the high-significance gravitational wave (GW) signal S190814bv. The GW data indicated that the event resulted from a neutron star–black hole (NSBH) merger, or potentially a low-mass binary BH merger. Due to the low false-alarm rate and the precise localization (23 deg 2 at 90%), S190814bv presented the community with the best opportunity yet to directly observe an optical/near-infrared counterpart to an NSBH merger. To search for potential counterparts, the GROWTH Collaboration performed real-time image subtraction on six nights of public Dark Energy Camera images acquired in the 3 weeks following the merger, covering >98% of the localization probability. Using a worldwide network of follow-up facilities, we systematically undertook spectroscopy and imaging of optical counterpart candidates. Combining these data with a photometric redshift catalog, we ruled out each candidate as the counterpart to S190814bv and placed deep, uniform limits on the optical emission associated with S190814bv. For the nearest consistent GW distance, radiative transfer simulations of NSBH mergers constrain the ejecta mass of S190814bv to be M ej < 0.04 M ⊙ at polar viewing angles, or M ej < 0.03 M ⊙ if the opacity is κ < 2 cm 2 g -1 . Assuming a tidal deformability for the NS at the high end of the range compatible with GW170817 results, our limits would constrain the BH spin component aligned with the orbital momentum to be χ < 0.7 for mass ratios Q < 6, with weaker constraints for more compact NSs.

79 ASTRONOMY AND ASTROPHYSICS↗

The transition regions of Capella

We have used the Goddard High Resolution Spectrometer (GHRS) to observe the spectoscopic binary system Capella (G8 III + G1 III). Exposures with the G140L, G140M, G160M, G200M, and echelle gratings provide emission line profiles with unprecedented signal-to-noise and spectral resolving power (lambda/Delta-lambda) up to 92,000. Multi-Gaussin fits to the line profiles show that the hotter star contributes 60%-70% of the total flux in the chromospheric O I and Mg II resonance lines, but about 90% of the flux in the Si III, Si IV, and C IV lines formed in the transition region at T less than or = 10(exp 5) K. We find clear evidence that the emission lines from the hotter star are systemtically redshifted relative to the photosphere with Doppler shifts of 5 +/- 1 km/s for the +9 +/- 3 km/s in the chromospheric Mg II and O I lines, respectively, increasing to +24 +/- 5 km/s for the transition region Si IV 1393.8A line. The multi-Gaussian fits to permitted transition region lines of SI III, Si IV, C IV, and N V indicate the presence of three components: moderately broad lines formed in the transition region of the hotter star (component H), narrow lines formed in the transition region of the cooler star (component C), and very broad lines that we think are formed in microflares on the hotter star (component B). The He II 1640.4 A feature has an broad profile, which indicates that it is formed by collisional excitation primarily from the hotter star, and a weak narrow component that we interpret as due to radiative recombination on the cooler star. We observed spin-forbidden emission lines of C III), O III), Si III), O IV), O V), and S IV) that are sensitive to electron density. Fainter members of the O IV) multiplet and all of the S IV) lines have never before been seen in any star than the Sun. We determine electron densities in the transition regions of the Capella stars using lines ratios of O IV) lines and emission measure analysis. The emission measures are self-consistent only when the fluxes from each emitting component are considered separately. In particular, the transition region abundance distributions appear to be different on the two stars, and the spin-forbidden lines were not detected in the B component. The emission measures for component H (the G1 star) are consistent with a constant electron pressure (or hydrostatic equilibrium) transition region with P(sub e) = 10(exp 15 + 0.1)/cu cm K and possibly solar 'coronal' abundances.

Linsky, Jeffrey L.↗

Impact of Mergers on USA Parameter Estimation for Nonspinning Black Hole Binaries

We investigate the precision with which the parameters describing the characteristics and location of nonspinning black hole binaries can be measured with the Laser Interferometer Space Antenna (LISA). By using complete waveforms including the inspiral, merger and ringdown portions of the signals, we find that LISA will have far greater precision than previous estimates for nonspinning mergers that ignored the merger and ringdown. Our analysis covers nonspinning waveforms with moderate mass ratios, q > or = 1/10, and total masses 10(exp 5) < M/M_{Sun} < 10(exp 7). We compare the parameter uncertainties using the Fisher matrix formalism, and establish the significance of mass asymmetry and higher-order content to the predicted parameter uncertainties resulting from inclusion of the merger. In real-time observations, the later parts of the signal lead to significant improvements in sky-position precision in the last hours and even the final minutes of observation. For comparable mass systems with total mass M/M_{Sun} = approx. 10(exp 6), we find that the increased precision resulting from including the merger is comparable to the increase in signal-to-noise ratio. For the most precise systems under investigation, half can be localized to within O(10 arcmin), and 18% can be localized to within O(1 arcmin).

McWilliams, Sean T.↗

GROWTH on S190814bv: Deep Synoptic Limits on the Optical/Near-infrared Counterpart to a Neutron Star–Black Hole Merger

On 2019 August 14, the Advanced LIGO and Virgo interferometers detected the high-significance gravitational wave (GW) signal S190814bv. The GW data indicated that the event resulted from a neutron star–black hole (NSBH) merger, or potentially a low-mass binary BH merger. Due to the low false-alarm rate and the precise localization (23 deg^(2) at 90%), S190814bv presented the community with the best opportunity yet to directly observe an optical/near-infrared counterpart to an NSBH merger. To search for potential counterparts, the GROWTH Collaboration performed real-time image subtraction on six nights of public Dark Energy Camera images acquired in the 3 weeks following the merger, covering >98% of the localization probability. Using a worldwide network of follow-up facilities, we systematically undertook spectroscopy and imaging of optical counterpart candidates. Combining these data with a photometric redshift catalog, we ruled out each candidate as the counterpart to S190814bv and placed deep, uniform limits on the optical emission associated with S190814bv. For the nearest consistent GW distance, radiative transfer simulations of NSBH mergers constrain the ejecta mass of S190814bv to be M(ej) < 0.04 Mꙩ at polar viewing angles, or M(ej) < 0.03 Mꙩ if the opacity is κ < 2 sq.cm/g. Assuming a tidal deformability for the NS at the high end of the range compatible with GW170817 results, our limits would constrain the BH spin component aligned with the orbital momentum to be χ < 0.7 for mass ratios Q < 6, with weaker constraints for more compact NSs.

Igor Andreoni↗

Search for subsolar-mass black hole binaries in the second part of Advanced LIGO’s and Advanced Virgo’s third observing run

ABSTRACT We describe a search for gravitational waves from compact binaries with at least one component with mass $0.2$–$1.0 \, \mathrm{M}_\odot$ and mass ratio q ≥ 0.1 in Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) and Advanced Virgo data collected between 2019 November 1, 15:00 utc and 2020 March 27, 17:00 utc. No signals were detected. The most significant candidate has a false alarm rate of $0.2 \, \rm {yr}^{-1}$. We estimate the sensitivity of our search over the entirety of Advanced LIGO’s and Advanced Virgo’s third observing run, and present the most stringent limits to date on the merger rate of binary black holes with at least one subsolar-mass component. We use the upper limits to constrain two fiducial scenarios that could produce subsolar-mass black holes: primordial black holes (PBH) and a model of dissipative dark matter. The PBH model uses recent prescriptions for the merger rate of PBH binaries that include a rate suppression factor to effectively account for PBH early binary disruptions. If the PBHs are monochromatically distributed, we can exclude a dark matter fraction in PBHs $f_\mathrm{PBH} \gtrsim \, 0.6$ (at 90 per cent confidence) in the probed subsolar-mass range. However, if we allow for broad PBH mass distributions, we are unable to rule out fPBH = 1. For the dissipative model, where the dark matter has chemistry that allows a small fraction to cool and collapse into black holes, we find an upper bound fDBH < 10−5 on the fraction of atomic dark matter collapsed into black holes.

79 ASTRONOMY AND ASTROPHYSICS↗

Computational Imaging for Intelligence in Highly Scattering Aerosols (Final Report)

Natural and man-made degraded visual environments pose major threats to national security. The random scattering and absorption of light by tiny particles suspended in the air reduces situational awareness and causes unacceptable down-time for critical systems and operations. To improve the situation, we have developed several approaches to interpret the information contained within scattered light to enhance sensing and imaging in scattering media. These approaches were tested at the Sandia National Laboratory Fog Chamber facility and with tabletop fog chambers. Computationally efficient light transport models were developed and leveraged for computational sensing. The models are based on a weak angular dependence approximation to the Boltzmann or radiative transfer equation that appears to be applicable in both the moderate and highly scattering regimes. After the new model was experimentally validated, statistical approaches for detection, localization, and imaging of objects hidden in fog were developed and demonstrated. A binary hypothesis test and the Neyman-Pearson lemma provided the highest theoretically possible probability of detection for a specified false alarm rate and signal-to-noise ratio. Maximum likelihood estimation allowed estimation of the fog optical properties as well as the position, size, and reflection coefficient of an object in fog. A computational dehazing approach was implemented to reduce the effects of scatter on images, making object features more readily discernible. We have developed, characterized, and deployed a new Tabletop Fog Chamber capable of repeatably generating multiple unique fog-analogues for optical testing in degraded visual environments. We characterized this chamber using both optical and microphysical techniques. In doing so we have explored the ability of droplet nucleation theory to describe the aerosols generated within the chamber, as well as Mie scattering theory to describe the attenuation of light by said aerosols, and correlated the aerosol microphysics to optical properties such as transmission and meteorological optical range (MOR). This chamber has proved highly valuable and has supported multiple efforts inclusive to and exclusive of this LDRD project to test optics in degraded visual environments. Circularly polarized light has been found to maintain its polarization state better than linearly polarized light when propagating through fog. This was demonstrated experimentally in both the visible and short-wave infrared (SWIR) by imaging targets made of different commercially available retroreflective films. It was found that active circularly polarized imaging can increase contrast and range compared to linearly polarized imaging. We have completed an initial investigation of the capability for machine learning methods to reduce the effects of light scattering when imaging through fog. Previously acquired experimental long-wave images were used to train an autoencoder denoising architecture. Overfitting was found to be a problem because of lack of variability in the object type in this data set. The lessons learned were used to collect a well labeled dataset with much more variability using the Tabletop Fog Chamber that will be available for future studies. We have developed several new sensing methods using speckle intensity correlations. First, the ability to image moving objects in fog was shown, establishing that our unique speckle imaging method can be implemented in dynamic scattering media. Second, the speckle decorrelation over time was found to be sensitive to fog composition, implying extensions to fog characterization. Third, the ability to distinguish macroscopically identical objects on a far-subwavelength scale was demonstrated, suggesting numerous applications ranging from nanoscale defect detection to security. Fourth, we have shown the capability to simultaneously image and localize hidden objects, allowing the speckle imaging method to be effective without prior object positional information. Finally, an interferometric effect was presented that illustrates a new approach for analyzing speckle intensity correlations that may lead to more effective ways to localize and image moving objects. All of these results represent significant developments that challenge the limits of the application of speckle imaging and open important application spaces. A theory was developed and simulations were performed to assess the potential transverse resolution benefit of relative motion in structured illumination for radar systems. Results for a simplified radar system model indicate that significant resolution benefits are possible using data from scanning a structured beam over the target, with the use of appropriate signal processing.

58 GEOSCIENCES↗

GW190814: Gravitational Waves from the Coalescence of a 23 Solar Mass Black Hole with a 2.6 Solar Mass Compact Object

We report the observation of a compact binary coalescence involving a 22.2-24.3 M ⊙ black hole and a compact object with a mass of 2.50-2.67 M ⊙ (all measurements quoted at the 90% credible level). The gravitational-wave signal, GW190814, was observed during LIGO's and Virgo's third observing run on 2019 August 14 at 21:10:39 UTC and has a signal-to-noise ratio of 25 in the three-detector network. The source was localized to 18.5 deg 2 at a distance of 241 +41-41 Mpc; no electromagnetic counterpart has been confirmed to date. The source has the most unequal mass ratio yet measured with gravitational waves , 0.112+0.0090.008, , and its secondary component is either the lightest black hole or the heaviest neutron star ever discovered in a double compact-object system. The dimensionless spin of the primary black hole is tightly constrained to ≤0.07. Tests of general relativity reveal no measurable deviations from the theory, and its prediction of higher-multipole emission is confirmed at high confidence. We estimate a merger rate density of 1-23 Gpc -3 yr -1 for the new class of binary coalescence sources that GW190814 represents. Astrophysical models predict that binaries with mass ratios similar to this event can form through several channels, but are unlikely to have formed in globular clusters. However, the combination of mass ratio, component masses, and the inferred merger rate for this event challenges all current models of the formation and mass distribution of compact-object binaries.

79 ASTRONOMY AND ASTROPHYSICS↗

Constraints on the Cosmic Expansion History from GWTC–3

We use 47 gravitational wave sources from the Third LIGO–Virgo–Kamioka Gravitational Wave Detector Gravitational Wave Transient Catalog (GWTC–3) to estimate the Hubble parameter H(z), including its current value, the Hubble constant H0. Each gravitational wave (GW) signal provides the luminosity distance to the source, and we estimate the corresponding redshift using two methods: the redshifted masses and a galaxy catalog. Using the binary black hole (BBH) redshifted masses, we simultaneously infer the source mass distribution and H(z). The source mass distribution displays a peak around 34 M ⊙ , followed by a drop-off. Assuming this mass scale does not evolve with the redshift results in a H(z) measurement, yielding H 0 = $68$$^{+12}_{–8}$km s –1 Mpc –1 (68% credible interval) when combined with the H 0 measurement from GW170817 and its electromagnetic counterpart. This represents an improvement of 17% with respect to the H 0 estimate from GWTC–1. The second method associates each GW event with its probable host galaxy in the catalog GLADE+, statistically marginalizing over the redshifts of each event's potential hosts. Assuming a fixed BBH population, we estimate a value of H 0 = $68$$^{+8}_{–6}$km s –1 Mpc –1 with the galaxy catalog method, an improvement of 42% with respect to our GWTC–1 result and 20% with respect to recent H 0 studies using GWTC–2 events. However, we show that this result is strongly impacted by assumptions about the BBH source mass distribution; the only event which is not strongly impacted by such assumptions (and is thus informative about H 0 ) is the well-localized event GW190814.

79 ASTRONOMY AND ASTROPHYSICS↗

Effect of atmospheric turbulence on the bit error probability of a space to ground near infrared laser communications link using binary pulse position modulation and an avalanche photodiode detector

The effect of atmospheric turbulence on the bit error rate of a space-to-ground near infrared laser communications link is investigated, for a link using binary pulse position modulation and an avalanche photodiode detector. Formulas are presented for the mean and variance of the bit error rate as a function of signal strength. Because these formulas require numerical integration, they are of limited practical use. Approximate formulas are derived which are easy to compute and sufficiently accurate for system feasibility studies, as shown by numerical comparison with the exact formulas. A very simple formula is derived for the bit error rate as a function of signal strength, which requires only the evaluation of an error function. It is shown by numerical calculations that, for realistic values of the system parameters, the increase in the bit error rate due to turbulence does not exceed about thirty percent for signal strengths of four hundred photons per bit or less. The increase in signal strength required to maintain an error rate of one in 10 million is about one or two tenths of a db.

Safren, H. G.↗