Holographic-(V)AE: an end-to-end SO(3)-Equivariant (Variational) Autoencoder in Fourier Space

Group-equivariant neural networks have emerged as a data-efficient approach to solve classification and regression tasks, while respecting the relevant symmetries of the data. However, little work has been done to extend this paradigm to the unsupervised and generative domains. Here, we present Holographic-(V)AE (H-(V)AE), a fully end-to-end SO(3)-equivariant (variational) autoencoder in Fourier space, suitable for unsupervised learning and generation of data distributed around a specified origin. H-(V)AE is trained to reconstruct the spherical Fourier encoding of data, learning in the process a latent space with a maximally informative invariant embedding alongside an equivariant frame describing the orientation of the data. We extensively test the performance of H-(V)AE on diverse datasets and show that its latent space efficiently encodes the categorical features of spherical images and structural features of protein atomic environments. Our work can further be seen as a case study for equivariant modeling of a data distribution by reconstructing its Fourier encoding

Learning the shape of protein micro-environments with a holographic convolutional neural network

Proteins play a central role in biology from immune recognition to brain activity. While major advances in machine learning have improved our ability to predict protein structure from sequence, determining protein function from structure remains a major challenge. Here, we introduce Holographic Convolutional Neural Network (H-CNN) for proteins, which is a physically motivated machine learning approach to model amino acid preferences in protein structures. H-CNN reflects physical interactions in a protein structure and recapitulates the functional information stored in evolutionary data. H-CNN accurately predicts the impact of mutations on protein function, including stability and binding of protein complexes. Our interpretable computational model for protein structure-function maps could guide design of novel proteins with desired function

Preliminary Spectral Analysis of the Type II Supernova 1999em

We have calculated fast direct spectral model fits to two early-time spectra of the Type-II plateau SN 1999em, using the SYNOW synthetic spectrum code. The first is an extremely early blue optical spectrum and the second a combined HST and optical spectrum obtained one week later. Spectroscopically this supernova appears to be a normal Type II and these fits are in excellent agreement with the observed spectra. Our direct analysis suggests the presence of enhanced nitrogen. We have further studied these spectra with the full NLTE general model atmosphere code PHOENIX. While we do not find confirmation for enhanced nitrogen (nor do we rule it out), we do require enhanced helium. An even more intriguing possible line identification is complicated Balmer and He I lines, which we show falls naturally out of the detailed calculations with a shallow density gradient. We also show that very early spectra such as those presented here combined with sophisticated spectral modeling allows an independent estimate of the total reddening to the supernova, since when the spectrum is very blue, dereddening leads to changes in the blue flux that cannot be reproduced by altering the ``temperature'' of the emitted radiation. These results are extremely encouraging since they imply that detailed modeling of early spectra can shed light on both the abundances and total extinction of SNe II, the latter improving their utility and reliability as distance indicators.Comment: to appear in ApJ, 2000, 54

Analysis of the Type IIn Supernova 1998S: Effects of Circumstellar Interaction on Observed Spectra

We present spectral analysis of early observations of the Type IIn supernova 1998S using the general non-local thermodynamic equilibrium atmosphere code \tt PHOENIX}. We model both the underlying supernova spectrum and the overlying circumstellar interaction region and produce spectra in good agreement with observations. The early spectra are well fit by lines produced primarily in the circumstellar region itself, and later spectra are due primarily to the supernova ejecta. Intermediate spectra are affected by both regions. A mass-loss rate of order $\dot M \sim 0.0001-0.001$\msol yr$^{-1}$ is inferred for a wind speed of 100-1000 \kmps. We discuss how future self-consistent models will better clarify the underlying progenitor structure.Comment: to appear in ApJ, 2001, 54

Coronal emission from the shocked circumstellar ring of SN 1987A

High resolution spectra with UVES/VLT of SN 1987A from December 2000 until November 2005 show a number of high ionization lines from gas with velocities of roughly 350 km/s, emerging from the shocked gas formed by the ejecta-ring collision. These include coronal lines from [Fe X], [Fe XI] and [Fe XIV] which have increased by a factor of about 20 during the observed period. The evolution of the lines is similar to that of the soft X-rays, indicating that they arise in the same component. The line ratios are consistent with those expected from radiative shocks with velocity 310-390 km/s, corresponding to a shock temperature of (1.6-2.5) x 10^6 K. A fraction of the coronal emission may, however, originate in higher velocity adiabatic shocks.Comment: 11 pages, 10 figures, accepted for publication in A&

Evolution of the Chandra CCD Spectra of SNR 1987A: Probing the Reflected-Shock Picture

We continue to explore the validity of the reflected shock structure (RSS) picture in SNR 1987A that was proposed in our previous analyses of the X-ray emission from this object. We used an improved version of our RSS model in a global analysis of 14 CCD spectra from the monitoring program with Chandra. In the framework of the RSS picture, we are able to match both the expansion velocity curve deduced from the analysis of the X-ray images and light curve. Using a simplified analysis, we also show that the X-rays and the non-thermal radio emission may originate from the same shock structure (the blast wave). We believe that using the RSS model in the analysis of grating data from the Chandra monitoring program of SNR 1987A that cover a long enough time interval, will allow us to build a more realistic physical picture and model of SNR 1987A.Comment: 14 pages, 1 Table, 8 figures, accepted for publication in MNRA

Hubble space telescope observations of high-velocity Lyα and Hα emission from supernova remnant 1987A: The structure and development of the reverse shock

We present two-dimensional line profiles of high-velocity (∼±12,000 km s -1) Lyα and Hα emission from supernova remnant 1987A obtained with the Space Telescope Imaging Spectrograph between 1997 September and 2001 September (days 3869-5327 after the explosion). This emission comes from hydrogen in the debris that is excited and ionized as it passes through the remnant's reverse shock. We use these profiles to measure the geometry and development of the reverse-shock surface. The observed emission is confined within ∼±30° about the remnant's equatorial plane. At the equator, the reverse shock has a radius of ∼75% of the distance to the equatorial ring. We detect marginal differences (6% ± 3%) between the location of the reverse-shock front in the northeast and southwest parts of the remnant. The radius of the reverse shock surface increases for latitudes above the equator, a geometry consistent with a model in which the supernova debris expands into a bipolar nebula. Assuming that the outer supernova debris has a power-law density distribution, we can infer from the reverse-shock emission light curve an expansion rate (in the northeast part of the remnant) of 3700 ± 900 km s -1, consistent with the expansion velocities determined from observations in radio (Manchester et al.) and X-ray (Park et al.; Michael et al.) wavelengths. However, our most recent observation (at day 5327) suggests that the rate of increase of mass flux across the northeast sector of the reverse shock has accelerated, perhaps because of deceleration of the reverse shock caused by the arrival of a reflected shock created when the blast wave struck the inner ring. Resonant scattering within the supernova debris causes Lyα photons created at the reverse shock to be directed preferentially outward, resulting in a factor of ∼5 difference in the observed brightness of the reverse shock in Lyαa between the near and far sides of the remnant. Accounting for this effect, we compare the observed reverse-shock Lyα and Hα fluxes to infer the amount of interstellar extinction by dust as E(B - V) = 0.17 ± 0.01 mag. We also notice extinction by dust in the equatorial ring with E(B - V) ≈ 0.02-0.08 mag, which implies dust-to-gas ratios similar to that of the LMC. Since Hα photons are optically thin to scattering, the observed asymmetry in brightness of Hα from the near and far sides of the remnant represents a real asymmetry in the mass flux through the reverse shock of ∼30%. We discuss future observational strategies that will permit us to further investigate the reverse-shock dynamics and resonant scattering of the Lyα line and to constrain better the extinction by dust within and in front of the remnant.published_or_final_versio

Modeling the Hubble Space Telescope ultraviolet and optical spectrum of spot 1 on the circumstellar ring of SN 1987A

We report and interpret Hubble Space Telescope (HST) Space Telescope Imaging Spectrograph (STIS) long-slit observations of the optical and ultraviolet (1150-10270 Å) emission line spectra of the rapidly brightening spot 1 on the equatorial ring of SN 1987A between 1997 September and 1999 October (days 3869-4606 after outburst). The emission is caused by radiative shocks created where the supernova blast wave strikes dense gas protruding inward from the equatorial ring. We measure and tabulate line identifications, fluxes, and, in some cases, line widths and shifts. We compute flux correction factors to account for substantial interstellar line absorption of several emission lines. Nebular analysis shows that optical emission lines come from a region of cool (T e ≈ 10 4 K) and dense (n e ≈ 10 6 cm -3) gas in the compressed photoionized layer behind the radiative shock. The observed line widths indicate that only shocks with shock velocities V s < 250 km s -1 have become radiative, while line ratios indicate that much of the emission must have come from yet slower (V s ≲ 135 km s -1) shocks. Such slow shocks can be present only if the protrusion has atomic density n ≳ 3 × 10 4 cm -3, somewhat higher than that of the circumstellar ring. We are able to fit the UV fluxes with an idealized radiative shock model consisting of two shocks (V s = 135 and 250 km s -1). The observed UV flux increase with time can be explained by the increase in shock surface areas as the blast wave overtakes more of the protrusion. The observed flux ratios of optical to highly ionized UV lines are greater by a factor of ∼2-3 than predictions from the radiative shock models, and we discuss the possible causes. We also present models for the observed Ha line widths and profiles, which suggest that a chaotic flow exists in the photoionized regions of these shocks. We discuss what can be learned with future observations of all the spots present on the equatorial ring.published_or_final_versio

Modelling the Hubble Space Telescope Ultraviolet and Optical Spectrum of Spot 1 on the Circumstellar Ring of SN 1987A

We report and interpret HST/STIS long-slit observations of the optical and ultraviolet (1150 - 10270 Angstrom) emission-line spectra of the rapidly brightening Spot 1 on the equatorial ring of SN 1987A between 1997 September and 1999 October (days 3869 -- 4606 after outburst). The emission is caused by radiative shocks created where the supernova blast wave strikes dense gas protruding inward from the equatorial ring. We measure and tabulate line identifications, fluxes and, in some cases, line widths and shifts. We compute flux correction factors to account for substantial interstellar line absorption of several emission lines. Nebular analysis shows that optical emission lines come from a region of cool (T_e ~ 10^4 K) and dense (n_e ~ 10^6 cm^-3) gas in the compressed photoionized layer behind the radiative shock. The observed line widths indicate that only shocks with shock velocities V_s < 250 km/s have become radiative, while line ratios indicate that much of the emission must have come from yet slower (V_s < 135 k/ms) shocks. We are able to fit the UV fluxes with an idealized radiative shock model consisting of two shocks (V_s = 135 and 250 km/s). The observed UV flux increase with time can be explained by the increase in shock surface areas as the blast wave overtakes more of the protrusion. The observed flux ratios of optical to highly-ionized UV lines are greater by a factor of ~ 2 -- 3 than predictions from the radiative shock models and we discuss the possible causes. We also present models for the observed H-alpha line widths and profiles, which suggests that a chaotic flow exists in the photoionized regions of these shocks. We discuss what can be learned with future observations of all the spots present on the equatorial ring

Monitoring the Evolution of the X-ray Remnant of SN 1987A

We report on the results of our monitoring program of the remnant of SN 1987A with the Advanced CCD Imaging Spectrometer (ACIS) on board the {\it Chandra X-ray Observatory}. Two new observations have been performed in AO2, bringing the total to four monitoring observations over the past two years. Over this time period, new techniques for correction of ``Charge Transfer Inefficiency (CTI)'' and for use of charge spreading to provide angular resolution somewhat better than the pixel size of the CCD detector have become available at Penn State. We have processed all four observations using sub-pixel resolution to obtain the highest possible angular resolution, and using our CTI correction software to provide more reliable spectral analysis and flux estimations. The high angular resolution images indicate that the X-ray bright knots are convincingly correlated with the optical spots, primarily at \la1 keV, while higher energy photons are very well correlated with radio images. Our data also provide marginal evidence for radial expansion of the X-ray remnant at a rate of 5200 $\pm$ 2100 km s$^{-1}$. The X-ray flux appears to linearly increase by $\sim$60% over the 18 month period of these observations. The spectrum is dominated by broad complexes of atomic emission lines and can be fit with a simple model of a plane-parallel shock with electron temperatures of $kT$ $\sim$ 2 $-$ 4 keV and a postshock electron density of $n_{e}$ $\sim$ 210 $-$ 420 cm$^{-3}$. The implied 0.5 $-$ 10 keV band luminosity in 2001 April is $\sim$1.3 $\times$ 10$^{35}$ ergs s$^{-1}$; as of that date, we still observe no direct evidence for the central point source, with an upper limit on the {\it observed} luminosity of $L_{X}$ $\sim$ 5.5 $\times$ 10$^{33}$ ergs s$^{-1}$ in the 2 $-$ 10 keV band.Comment: 29 pages, 5 figures (2 color images), Accepted for Ap