Constant Spectral Index Research Articles

Sky-averaged 21-cm signal extraction using multiple antennas with an SVD framework: the REACH case

ABSTRACT In a sky-averaged 21-cm signal experiment, the uncertainty of the extracted signal depends mainly on the covariance between the foreground and 21-cm signal models. In this paper, we construct these models using the modes of variation obtained from the singular value decomposition of a set of simulated foreground and 21-cm signals. We present a strategy to reduce this overlap between the 21-cm and foreground modes by simultaneously fitting the spectra from multiple different antennas, which can be used in combination with the method of utilizing the time dependence of foregrounds while fitting multiple drift scan spectra. To demonstrate this idea, we consider two different foreground models (i) a simple foreground model, where we assume a constant spectral index over the sky, and (ii) a more realistic foreground model, with a spatial variation of the spectral index. For the simple foreground model, with just a single antenna design, we are able to extract the signal with good accuracy if we simultaneously fit the data from multiple time slices. The 21-cm signal extraction is further improved when we simultaneously fit the data from different antennas as well. This improvement becomes more pronounced while using the more realistic mock observations generated from the detailed foreground model. We find that even if we fit multiple time slices, the recovered signal is biased and inaccurate for a single antenna. However, simultaneously fitting the data from different antennas reduces the bias and the uncertainty by a factor of 2–3 on the extracted 21-cm signal.

Particle re-acceleration and diffuse radio sources in the galaxy cluster Abell 1550

Context. Radio observations of galaxy clusters reveal a plethora of diffuse, steep-spectrum sources related to the re-acceleration of cosmic-ray electrons, such as halos, relics, and phoenices. In this context, the LOw Frequency ARray Low-Band Antenna (LOFAR-LBA) Sky Survey (LoLSS) provides the most sensitive images of the sky at 54 MHz to date, allowing us to investigate re-acceleration processes in a poorly explored frequency regime. Aims. We study diffuse radio emission in the galaxy cluster Abell 1550, with the aim of constraining particle re-acceleration in the intra-cluster medium. Methods. We exploited observations at four different radio frequencies: 54, 144, 400, and 1400 MHz. To complement our analysis, we made use of archival Chandra X-ray data. Results. At all frequencies we detect an ultra-steep spectrum radio halo (Sν ∝ ν−1.6) with an extent of ∼1.2 Mpc at 54 MHz. Its morphology follows the distribution of the thermal intra-cluster medium inferred from the Chandra observation. West of the centrally located head-tail radio galaxy, we detect a radio relic with a projected extent of ∼500 kpc. From the relic, a ∼600 kpc long bridge departs and connects with the halo. Between the relic and the radio galaxy, we observe what is most likely a radio phoenix, given its curved spectrum. The phoenix is connected to the tail of the radio galaxy through two arms, which show a nearly constant spectral index for ∼300 kpc. Conclusions. The halo could be produced by turbulence induced by a major merger, with the merger axis lying in the NE-SW direction. This is supported by the position of the relic, whose origin could be attributed to a shock propagating along the merger axis. It is possible that the same shock has also produced the phoenix through adiabatic compression, while we propose that the bridge could be generated by electrons which were pre-accelerated by the shock, and then re-accelerated by turbulence. Finally, we detect hints of gentle re-energisation in the two arms that depart from the tail of the radio galaxy.

A Bayesian ILC Method for CMB B-mode Posterior Estimation and Reconstruction of Primordial Gravity Wave Signal

Abstract The cosmic microwave background (CMB) radiation B-mode polarization signal contains the unique signature of primordial metric perturbations produced during the inflation. The separation of the weak CMB B-mode signal from strong foreground contamination in observed maps is a complex task, and proposed new-generation low-noise satellite missions compete with the weak signal level of this gravitational background. In this paper, for the first time, we employ a foreground model-independent internal linear combination (ILC) method to reconstruct the CMB B-mode signal using simulated observations over large angular scales of the sky of six frequency bands of the future-generation CMB mission Probe of Inflation and Cosmic Origins (PICO). We estimate the joint CMB B-mode posterior density following the interleaving Gibbs steps of the B-mode angular power spectrum and cleaned map samples using the ILC method. We extend and improve the earlier reported Bayesian ILC method to analyze weak CMB B-mode reconstruction by introducing noise-bias corrections at two stages during the ILC weight estimation. By performing 200 Monte Carlo simulations of the Bayesian ILC method, we find that our method can reconstruct the CMB signals and the joint posterior density accurately over large angular scales of the sky. We estimate the Blackwell–Rao statistics of the marginal density of the CMB B-mode angular power spectrum and use them to estimate the joint density of the tensor-to-scalar ratio r and a lensing power spectrum amplitude A lens . Using 200 Monte Carlo simulations of the delensing approach, we find that our method can achieve an unbiased detection of the primordial gravitational-wave signal r with more than 8σ significance for levels of r ≥ 0.01 for input frequency maps with realistic lensing and foregrounds with constant spectral indices.

A thousand days after the merger: Continued X-ray emission from GW170817

ABSTRACT Recent observations with the Chandra X-ray telescope continue to detect X-ray emission from the transient GW170817. In a total exposure of 96.6 ks, performed between 2020 March 9 and 16 (935–942 d after the merger), a total of 8 photons are measured at the source position, corresponding to a significance of ≈5σ. Radio monitoring with the Australian Telescope Compact Array (ATCA) shows instead that the source has faded below our detection threshold (<33 $\mu$Jy, 3σ). By assuming a constant spectral index of β = 0.585, we derive an unabsorbed X-ray flux of ≈1.4 × 10−15 erg cm−2 s−1, higher than earlier predictions, yet still consistent with a simple structured jet model. We discuss possible scenarios that could account for prolonged emission in X-rays. The current data set appears consistent both with energy injection by a long-lived central engine and with the onset of a kilonova afterglow, arising from the interaction of the sub-relativistic merger ejecta with the surrounding medium. Long-term monitoring of this source will be essential to test these different models.

3D MHD simulations and synthetic radio emission from an oblique rotating magnetic massive star

ABSTRACT We have performed 3D isothermal MHD simulation of a magnetic rotating massive star with a non-zero dipole obliquity and predicted the radio/sub-mm observable light curves and continuum spectra for a frequency range compatible with ALMA. From these results we also compare the model input mass-loss to that calculated from the synthetic thermal emission. Spherical and cylindrical symmetry is broken due to the obliquity of the stellar magnetic dipole resulting in an inclination and phase dependence of both the spectral flux and inferred mass-loss rate, providing testable predictions of variability for oblique rotator. Both quantities vary by factors between 2 and 3 over a full rotational period of the star, demonstrating that the role of rotation as critical in understanding the emission. This illustrates the divergence from a symmetric wind, resulting in a two-armed spiral structure indicative of an oblique magnetic rotator. We show that a constant spectral index, α, model agrees well with our numerical prediction for a spherical wind for ν < 103 GHz; however it is unable to capture the behaviour of emission at ν > 103 GHz. As such we caution the use of such constant α models for predicting emission from non-spherical winds such as those which form around magnetic massive stars.

Far-infrared photometry of OJ 287 with the Herschel Space Observatory

Context. The blazar OJ 287 has shown a ≈12 year quasi-periodicity over more than a century, in addition to the common properties of violent variability in all frequency ranges. It is the strongest known candidate to have a binary singularity in its central engine. Aim. We aim to better understand the different emission components by searching for correlated variability in the flux over four decades of frequency measurements. Methods. We combined data at frequencies from the millimetric to the visible to characterise the multifrequency light curve in April and May 2010. This includes the only photometric observations of OJ 287 made with the Herschel Space Observatory: five epochs of data obtained over 33 days at 250, 350, and 500 μm with Herschel-SPIRE. Results. Although we find that the variability at 37 GHz on timescales of a few weeks correlates with the visible to near-IR spectral energy distribution, there is a small degree of reddening in the continuum at lower flux levels that is revealed by the decreasing rate of decline in the light curve at lower frequencies. However, we see no clear evidence that a rapid flare detected in the light curve during our monitoring in the visible to near-IR light curve is seen either in the Herschel data or at 37 GHz, suggesting a low-frequency cut-off in the spectrum of such flares. Conclusions.We see only marginal evidence of variability in the observations with Herschel over a month, although this may be principally due to the poor sampling. The spectral energy distribution between 37 GHz and the visible can be characterised by two components of approximately constant spectral index: a visible to far-IR component of spectral index α = −0.95, and a far-IR to millimetric spectral index of α = −0.43. There is no evidence of an excess of emission that would be consistent with the 60 μmdust bump found in many active galactic nuclei.

Can power spectrum observations rule out slow-roll inflation?

The spectral index of scalar perturbations is an important observable that allows us to learn about inflationary physics. In particular, a detection of a significant deviation from a constant spectral index could enable us to rule out the simplest class of inflation models. We investigate whether future observations could rule out canonical single-field slow-roll inflation given the parameters allowed by current observational constraints. We find that future measurements of a constant running (or running of the running) of the spectral index over currently available scales are unlikely to achieve this. However, there remains a large region of parameter space (especially when considering the running of the running) for falsifying the assumed class of slow-roll models if future observations accurately constrain a much wider range of scales.

A concordant scenario to explain FU Orionis from deep centimeter and millimeter interferometric observations

We resolved FU Ori at 29-37 GHz using the JVLA with $\sim$0$''$.07 resolution, and performed the complementary JVLA 8-10 GHz observations, the SMA 224 GHz and 272 GHz observations, and compared with archival ALMA 346 GHz observations to obtain the SEDs. Our 8-10 GHz observations do not find evidence for the presence of thermal radio jets, and constrain the radio jet/wind flux to at least 90 times lower than the expected value from the previously reported bolometric luminosity-radio luminosity correlation. The emission at $>$29 GHz may be dominated by the two spatially unresolved sources, which are located immediately around FU Ori and its companion FU Ori S, respectively. Their deconvolved radii at 33 GHz are only a few au. The 8-346 GHz SEDs of FU Ori and FU Ori S cannot be fit by constant spectral indices (over frequency). The more sophisticated models for SEDs suggest that the $>$29 GHz emission is contributed by a combination of free-free emission from ionized gas, and thermal emission from optically thick and optically thin dust components. We hypothesize that dust in the innermost parts of the disks ($\lesssim$0.1 au) has been sublimated, and thus the disks are no more well shielded against the ionizing photons. The estimated overall gas and dust mass based on SED modeling, can be as high as a fraction of a solar mass, which is adequate for developing disk gravitational instability. Our present explanation for the observational data is that the massive inflow of gas and dust due to disk gravitational instability or interaction with a companion/intruder, was piled up at the few au scale due to the development of a deadzone with negligible ionization. The piled up material subsequently triggered the thermal and the MRI instabilities when the ionization fraction in the inner sub-au scale region exceeded a threshold value, leading to the high protostellar accretion rate.

RADIO CONTINUUM EMISSION FROM THE MAGNETAR SGR J1745-2900: INTERACTION WITH GAS ORBITING Sgr A*

We present radio continuum light curves of the magnetar SGR J1745$-$2900 and Sgr A* obtained with multi-frequency, multi-epoch Very Large Array observations between 2012 and 2014. During this period, a powerful X-ray outburst from SGR J1745$-$2900 occurred on 2013-04-24. Enhanced radio emission is delayed with respect to the X-ray peak by about seven months. In addition, the flux density of the emission from the magnetar fluctuates by a factor of 2 to 4 at frequencies between 21 and 41 GHz and its spectral index varies erratically. Here we argue that the excess fluctuating emission from the magnetar arises from the interaction of a shock generated from the X-ray outburst with the orbiting ionized gas at the Galactic center. In this picture, variable synchrotron emission is produced by ram pressure variations due to inhomogeneities in the dense ionized medium of the Sgr A West bar. The pulsar with its high transverse velocity is moving through a highly blue-shifted ionized medium. This implies that the magnetar is at a projected distance of $\sim0.1$ pc from Sgr A* and that the orbiting ionized gas is partially or largely responsible for a large rotation measure detected toward the magnetar. Despite the variability of Sgr A* expected to be induced by the passage of the G2 cloud, monitoring data shows a constant flux density and spectral index during this period

New insights into the evolution of the FR I radio galaxy 3C 270 (NGC 4261) from VLA and GMRT radio observations

We present the Giant Metrewave Radio Telescope 240 MHz observations of the nearby luminous Fanaroff–Riley type I radio source 3C 270, in the group-central elliptical NGC 4261. Combining these data with reprocessed Very Large Array 1.55 and 4.8 GHz observations, we produce spectral-index maps that reveal a constant spectral index along the jets and a gradual steepening from the ends of the jets through the lobes towards the nucleus. A Jaffe & Perola (JP) model fitted to the integrated spectrum of the source gives an asymptotic low-frequency index of |$\alpha _{\rm {inj}}=0.53_{-0.02}^{+0.01}$|⁠, while JP models fitted to the observed spectral index trend along the lobes allow us to estimate radiative ages of ∼29 and ∼37 Myr for the west and east lobes, respectively. Our age estimates are a factor of 2 lower than the 75-Myr upper limit derived from X-ray data. We find unlikely the scenario of an early supersonic phase in which the lobe expanded into the ISM at approximately Mach 6 (3500 km s−1), and suggest that either the source underwent multiple active galactic nuclei outbursts with possible large changes in jet power, or possibly that the source age that we find is due to a backflow that transports young electrons from the jet tips through the lobes towards the nucleus relatively quickly. We calculate that in the lobes the energy ratio of non-radiating to radiating particles is ∼4–24 indicating significant gas entrainment. If the lobes are in pressure balance with their surroundings, the total energy required to heat the entrained material is 1058 erg, ∼40 per cent of the total enthalpy of the lobes.

Suppressing the impact of a high tensor-to-scalar ratio on the temperature anisotropies

The BICEP2 collaboration has reported a strong B mode signal in the CMB polarization, which is well fit by a tensor-to-scalar ratio of r ≃ 0.2. This is greater than the upper limit r < 0.11 obtained from the temperature anisotropies under the assumption of a constant scalar spectral index ns. This discrepancy can be reduced once the statistical error and the contamination from polarized dust are accounted for. If however a large value for r will be confirmed, it will need to be reconciled with the temperature anisotropies data. The most advocated explanation involves a variation of ns with scales (denoted as running) that has a magnitude significantly greater than the generic slow roll predictions. We instead study the possibility that the large scale temperature anisotropies are not enhanced because of a suppression of the scalar power at large scales. Such a situation can be achieved for instance by a sudden change of the speed of the inflaton (by about 14%), and we show that it fits the temperature anisotropies and polarization data considerably better than a constant running (its χ2 improves by ∼ 7.5 over that of the constant running, at the cost of one more parameter). We also consider the possibility that the large scale temperature fluctuations are suppressed by an anti-correlation between tensor and scalar modes. Unfortunately, while such effect does affect the temperature fluctuations at large scales, it does not affect the temperature power spectrum and cannot, therefore, help in reconciling a large value of r with the limits from temperature fluctuations.

Observations and analysis of phase scintillation of spacecraft signal on the interplanetary plasma

Aims. The phase scintillation of the European Space Agency's (ESA) Venus Express (VEX) spacecraft telemetry signal was observed at X-band (\lambda = 3.6 cm) with a number of radio telescopes of the European VLBI Network (EVN) in the period 2009-2013. Methods. We found a phase fluctuation spectrum along the Venus orbit with a nearly constant spectral index of -2.42 +/-0.25 over the full range of solar elongation angles from 0{\deg} to 45{\deg}, which is consistent with Kolmogorov turbulence. Radio astronomical observations of spacecraft signals within the solar system give a unique opportunity to study the temporal behaviour of the signal's phase fluctuations caused by its propagation through the interplanetary plasma and the Earth's ionosphere. This gives complementary data to the classical interplanetary scintillation (IPS) study based on observations of the flux variability of distant natural radio sources. Results. We present here our technique and the results on IPS. We compare these with the total electron content (TEC) for the line of sight through the solar wind. Finally, we evaluate the applicability of the presented technique to phase-referencing Very Long Baseline Interferometry (VLBI) and Doppler observations of currently operational and prospective space missions.

THE CONTRIBUTION OF RADIO GALAXY CONTAMINATION TO MEASUREMENTS OF THE SUNYAEV-ZEL'DOVICH DECREMENT IN MASSIVE GALAXY CLUSTERS AT 140 GHz WITH BOLOCAM

We describe in detail our characterization of the compact radio source population in 140 GHz Bolocam observations of a set of 45 massive galaxy clusters. We use a combination of 1.4 and 30 GHz data to select a total of 28 probable cluster-member radio galaxies and also to predict their 140 GHz flux densities. All of these galaxies are steep-spectrum radio sources and they are found preferentially in the cool-core clusters within our sample. In particular, 11 of the 12 brightest cluster member radio sources are associated with cool-core systems. Although none of the individual galaxies are robustly detected in the Bolocam data, the ensemble-average flux density at 140 GHz is consistent with, but slightly lower than, the extrapolation from lower frequencies assuming a constant spectral index. In addition, our data indicate an intrinsic scatter of 30 percent around the power-law extrapolated flux densities at 140 GHz, although our data do not tightly constrain this scatter. For our cluster sample, which is composed of high-mass and moderate-redshift systems, we find that the maximum fractional change in the Sunyaev-Zel'dovich signal integrated over any single cluster due to the presence of these radio sources is 20 percent, and only 1/4 of the clusters show a fractional change of more than 1 percent. The amount of contamination is strongly dependent on cluster morphology, and nearly all of the clusters with more than 1 percent contamination are cool-core systems. This result indicates that radio contamination is not significant compared to current noise levels in 140 GHz images of massive clusters and is in good agreement with the level of radio contamination found in previous results based on lower frequency data or simulations.

The JCMT Gould Belt Survey: SCUBA-2 observations of radiative feedback in NGC 1333

Abstract We present observations of NGC 1333 from SCUBA-2 on the James Clerk Maxwell Telescope (JCMT), observed as a JCMT Gould Belt Survey pilot project during the shared risk campaign when the first of four arrays was installed at each of 450 and 850 μm. Temperature maps are derived from 450 and 850 μm ratios under the assumption of constant dust opacity spectral index β = 1.8. Temperatures indicate that the dust in the northern (IRAS 6/8) region of NGC 1333 is hot, 20–40 K, due to heating by the B star SVS3, other young stars in the IR/optically visible cluster and embedded protostars. Other luminous protostars are also identified by temperature rises at the 17 arcsec resolution of the ratio maps (0.02 pc assuming a distance of 250 pc for Perseus). The extensive heating raises the possibility that the radiative feedback may lead to increased masses for the next generation of stars.

Spectrum and ionization rate of low-energy Galactic cosmic rays

Abstract We consider the rate of ionization of diffuse and molecular clouds in the interstellar medium by Galactic cosmic rays (GCRs) in order to constrain its low-energy spectrum. We extrapolate the GCR spectrum obtained from PAMELA at high energies (≥200 GeV nucleon−1) and a recently derived GCR proton flux at 1–200 GeV from observations of gamma-rays from molecular clouds, and find that the observed average Galactic ionization rate can be reconciled with this GCR spectrum if there is a low-energy cut-off for protons at 10–100 MeV. We also identify the flattening below a few GeV as being due to (a) decrease of the diffusion coefficient and dominance of convective loss at low energy and (b) the expected break in energy spectrum for a constant spectral index in momentum. We show that the inferred CR proton spectrum of Φ ∝ E−1.7±0.2kin for Ekin ≤ few GeV is consistent with a power-law spectrum in momentum p−2.45± 0.4, which we identify as the spectrum at source. Diffusion loss at higher energies then introduces a steepening by E−α with α ∼ 1/3, making it consistent with high-energy measurements.

Blazar synchrotron emission of instantaneously power-law injected electrons under linear synchrotron, non-linear SSC, and combined synchrotron-SSC cooling

The broadband SEDs of blazars show two distinct components which in leptonic models are associated with synchrotron and SSC emission of highly relativistic electrons. In some sources the SSC component dominates the synchrotron peak by one or more orders of magnitude implying that the electrons mainly cool by inverse Compton collisions with their self-made synchrotron photons. Therefore, the linear synchrotron loss of electrons, which is normally invoked in emission models, has to be replaced by a nonlinear loss rate depending on an energy integral of the electron distribution. This modified electron cooling changes significantly the emerging radiation spectra. It is the purpose of this work to apply this new cooling scenario to relativistic power-law distributed electrons, which are injected instantaneously into the jet. We will first solve the differential equation of the volume-averaged differential number density of the electrons, and then discuss their temporal evolution. Since any non-linear cooling will turn into linear cooling after some time, we also calculated the electron number density for a combined cooling scenario consisting of both the linear and non-linear cooling. For all cases, we will also calculate analytically the emerging optically thin synchrotron fluence spectrum which will be compared to a numerical solution. For small normalized frequencies f < 1 the fluence spectra show constant spectral indices. We find for linear cooling a_SYN = 1/2, and for non-linear cooling a_SSC = 3/2. In the combined cooling scenario we obtain for the small injection parameter b_1 = 1/2, and for the large injection parameter b_2 = 3/2, which becomes b_1 = 1/2 for very small frequencies, again. This is the same behaviour as for monoenergetically injected electrons.

Extreme flux states of NGC 4151 observed with INTEGRAL★

We present a comprehensive spectral analysis of all INTEGRAL data obtained so far for the X-ray--bright Seyfert galaxy NGC 4151. We also use all contemporaneous data from RXTE, XMM, Swift and Suzaku. We find a linear correlation between the medium and hard-energy X-ray fluxes measured by INTEGRAL, which indicates an almost constant spectral index over six years. The majority of INTEGRAL observations were made when the source was either at a very bright or very dim hard--X-ray state. We find that thermal Comptonization models applied to the bright state yields the plasma temperature of 50--70 keV and its optical depth of 1.3--2.6, depending on the assumed source geometry. For the dim state, these parameters are in the ranges of 180--230 keV and 0.3--0.7, respectively. The Compton parameter is y = 1 for all the spectra, indicating a stable geometry. Using this result, we can determine the reflection effective solid angles associated with the close and distant reprocessing media as = 0.3 x 2pi and 0.2 x 2pi, respectively. The plasma energy balance, the weak disc reflection and a comparison of the UV fluxes illuminating the plasma to the observed ones are all consistent with an inner hot accretion surrounded by an outer cold disc. The disc truncation radius can be determined from an approximate equipartition between the observed UV and X-ray emission, and from the fitted disc blackbody model, as 15 gravitational radii. Alternatively, our results can be explained by a mildly relativistic coronal outflow.

Primordial “fNL” non-Gaussianity and perturbations beyond the present horizon

We show a primordial nonlinear (``${f}_{\mathrm{NL}}$'') term may produce unphysically large CMB anisotropy for a red-tilted primordial power spectrum ($n&lt;1$) because of coupling to primordial fluctuation on the largest scale. We consider a primordial power spectrum models of a running spectral index, and a transition at very low wave numbers. We find that only negative running spectral index models are allowed, provided that there is no transition at a low wave numbers (i.e. $k\ensuremath{\ll}1$). For models of a constant spectral index, we find $\mathrm{log}({k}_{c}/{k}_{0})\ensuremath{\gtrsim}\ensuremath{-}184$, at $1\ensuremath{\sigma}$ level, on the transition scale of sharp cutoff models, using recent CMB and SDSS data.

Consequences of the common origin of the knee and ankle in Cosmic Ray Physics

The differential energy spectrum of the cosmic radiation from solar modulation energies up to 5x10**19 eV is correctly predicted by a recent theory of the knee and ankle which uses only one normalization point. This remarkable quantitative result, spanning over many decades in energy and intensity, along with the existence of the second knee at 6x10**17 eV, is obtained assuming constant spectral indices of individual ions at the cosmic-ray sources and no other critical hypotheses. In this study the chemical composition of the cosmic radiation is evaluated as a direct consequence of the theory. The computed mean logarithmic mass exhibits a rising trend from 1.8 to 3.0 in the range 10**15-10**17 eV, a maximum value of 3.2 at 3x10**17 eV, and a characteristic lightening above 3x10**17 eV up to 4x10**18 eV. All of these distinctive features are in accord with the data of many experiments. Two additional consequences intrinsic to the theory are qualitatively discussed: (1) some limitative bounds on the mechanism accelerating cosmic rays; (2) the degree of isotropy implied by the residence time of the cosmic rays in the Galaxy.

Fast spectral fitting of hard X-ray bremsstrahlung from truncated power-law electron spectra

Hard X-Ray bremsstrahlung continuum spectra, such as from solar flares, are commonly described in terms of power-law fits, either to the photon spectra themselves or to the electron spectra responsible for them. In applications various approximate relations between electron and photon spectral indices are often used for energies both above and below electron low-energy cutoffs. We examine the form of the exact relationships in various situations, and for various cross-sections, showing that empirical relations sometimes used can be highly misleading and consider how to improve fitting procedures. We obtain expressions for photon spectra from single, double and truncated power-law electron spectra for a variety of cross-sections and for the thin and thick target models and simple analytic expressions for the Bethe-Heitler cases. We show that above a low-energy cutoff the Kramers and Bethe-Heitler results match reasonably well with results for exact cross-sections up to energies around 100 keV; that below the low-energy cutoff, Kramers and other constant spectral index forms commonly used are very poor approximations to accurate results; but that our analytical forms are a very good match. Analytical forms of the Bethe-Heitler photon spectra from general power-law electron spectra are an excellent match to exact results for both thin and thick targets and they enable much faster spectral fitting than evaluation of the full spectral integrations.