Past exam of the frontiers of Observational Astrophysics course of the University of Cambridge / 2015 / Paper 1 / 3

Stellar reflex motion. A planet and its host orbit their common barycentre. The star's line-of-sight velocity shifts its absorption lines by $\Delta \lambda /\lambda \simeq v_r/c$ at low speeds. A time series can be fitted withHere $\nu (t)$ is true anomaly, $\varpi$ the argument of periastron and $\gamma$ an arbitrary systemic velocity offset. At small companion mass and known stellar mass, the orbit supplies $M_p\sin i$, not the true planet mass without an inclination constraint. Long monitoring helps distinguish a reproducible Keplerian signal from stellar oscillations, magnetic activity, spots and instrumental trends.

The Lick reference-cell implementation. Starlight passes through a temperature-controlled cell of molecular iodine before entering the Hamilton high-resolution echelle spectrograph. The cell imprints a dense pattern of reference absorption lines on the stellar spectrum, predominantly in the approximately $500$–$630\mathrm{nm}$ interval. These lines share the same exposure and downstream optical path as the stellar light. A calibrated high-resolution Fourier-transform iodine spectrum and an iodine-free stellar template allow the wavelength solution and instrumental line-spread function to be fitted along with the stellar Doppler shift. The method corrects shifts and line-shape changes that a separate calibration exposure would not necessarily follow.^[ref]

A useful forward model for one short spectral segment isfollowed by pixel integration, continuum scaling and a fitted wavelength calibration. Here $S$ is the intrinsic stellar spectrum, $T_{{\mathrm{I}}_2}$ the cell transmission and $\mathcal{P}$ the instrumental line-spread function. The spectra multiply before convolution; simply dividing a recorded blended spectrum by the laboratory iodine spectrum is not equivalent to this model. The iodine pattern is stationary in the observatory frame while stellar lines move relative to it. Separate spectral segments accommodate wavelength dependence of the instrumental profile.

Earth's rotational and orbital velocities must then be removed with a barycentric correction using accurate timing, ideally weighted by photon arrival rate. Many stellar lines constrain a shift smaller than one instrumental resolution element; spectral resolution alone is not the velocity error. Photon statistics, template quality, instrumental-profile modelling and intrinsic stellar variability limit the precision. The method measures changes relative to a template and does not by itself provide a perfectly absolute centre-of-mass velocity. No particular discovery, modern precision floor or historical observing performance is assumed here.

Atmospheric turbulence produces time-dependent refractive-index fluctuations and phase errors across the telescope pupil. A large aperture's long-exposure point spread function is broadened approximately to $0.98\lambda /r_0$, where $r_0$ is the Fried parameter, rather than the diffraction scale $\lambda /D$. For fixed turbulence, $r_0\propto {\lambda }^{6/5}$, so the seeing angle improves weakly as ${\lambda }^{-1/5}$. The atmospheric coherence time scales with $r_0$ divided by an effective wind speed. This temporal scale concerns changing turbulence, not the optical bandwidth's coherence time.

Reduce the disturbance. A good high-altitude site, favourable weather, observing near the zenith, thermal control and dome ventilation reduce atmospheric and local seeing. They improve the uncorrected image but do not remove all high-order wavefront distortion. Active optics slowly maintains the main mirror's figure and alignment; it is distinct from the rapid adaptive correction of atmospheric phase errors.

Correct the wavefront. Adaptive optics uses a nearby natural guide star or artificial laser guide star, a wavefront sensor, a real-time controller and a deformable mirror. A Shack-Hartmann sensor estimates local wavefront slopes; mirror actuators apply the opposite optical-path distortion fast enough to track the atmosphere. A separate fast steering element or the mirror can correct image motion. In an ideal well-corrected direction this approaches diffraction-limited resolution and increases the concentration of light in the central image.

Finite guide-star flux, actuator sampling, control latency and anisoplanatism leave residual errors. A laser beacon samples a finite-height cone rather than the full distant-source cylinder, and a conventional laser reference does not determine absolute tip-tilt, so a natural reference remains useful. Multiple guide stars and multiple conjugate mirrors can widen the corrected field, with extra sensing and reconstruction complexity. These are fundamental geometry and control limits, not claims of a universal corrected sky fraction.^[ref]

Freeze and reconstruct. Short exposures can freeze the instantaneous speckle pattern. Speckle interferometry uses calibrated Fourier statistics to recover diffraction-scale structure; power-spectrum information alone loses phase, while bispectrum or other phase-sensitive measurements support image reconstruction. Lucky imaging selects sharp short frames, registers them and combines them, trading photon efficiency for improved image quality. Tip-tilt alignment alone removes image motion rather than all high-order aberration. These methods require sufficiently bright sources or references and can incur substantial read-noise penalties. Deconvolving a long-exposure image is useful only where a known point spread function and adequate signal preserve information; it cannot recover arbitrarily lost spatial frequencies. Moving to space avoids atmospheric seeing, although it changes the observing platform and its limitations rather than correcting a ground-based telescope.

Weak transitions in dilute gas. A forbidden emission line is a transition absent in the electric-dipole approximation because of its selection rules, but permitted through weaker channels such as magnetic-dipole or electric-quadrupole radiation, sometimes aided by state mixing. It is not an impossible transition. Its upper atomic level is relatively long lived, so collisions can deactivate it before a photon is emitted in dense gas. Low-density astrophysical plasmas allow radiative decay to compete successfully.

In a schematic two-level statistical equilibrium,The critical density is $n_{\mathrm{crit}}=A_{ul}/q_{ul}$ for this simple level system; multiple radiative and collisional channels require summed rates. At $n_e\ll n_{\mathrm{crit}}$ each collisional excitation is likely to produce radiation, so ${ϵ}_{ul}\propto n_l{n}_e{q}_{lu}h\nu$. At high density collisional de-excitation suppresses the line relative to this low-density expectation. The rate need not decrease absolutely with increasing density; the suppression is relative to the naive excitation scaling.

A temperature-sensitive ratio. For a Maxwell-Boltzmann electron distribution,where ${\rm Y}$ is an effective collision strength. Lines from upper levels with different excitation energies therefore have a strongly temperature-dependent ratio. Using one ion reduces sensitivity to abundance and ionization fraction compared with comparing unrelated species.

A standard example is doubly ionized oxygen: the $4959$ and $5007$ angstrom lines descend from ${}^1{D}_2$ toward the ${}^{3}P$ ground term, while the $4363$ angstrom line connects ${}^1{S}_0$ to ${}^1{D}_2$. The higher excitation threshold makes the auroral $4363$ line relatively stronger in hotter gas. In the low-density Maxwellian approximation, with slowly varying collision strengths,The numerical form is a useful approximation, not a universal exact atomic law; a multilevel atomic calculation with current collision strengths is needed for precise work.^[ref]

In an active galactic nucleus these lines commonly diagnose the relatively dilute narrow-line gas; dense broad-line gas can collisionally quench them. Measure and deblend the fluxes, subtract continuum and stellar features, correct differential extinction, and check density independently. The $5007/4959$ ratio mainly tests radiative branching and calibration because both lines share one upper level; it is not the required temperature thermometer. Temperature gradients, multiple density components, shocks, ionization structure and departures from the Maxwellian electron distribution can bias a single inferred temperature. Without a measured ratio and density regime, no particular AGN temperature is numerically determined here.

Measure counts in electrons so that photon and detector noise can be combined directly. For source electrons $S$, $n$ aperture pixels, total time $t$, sky rate $b$ per pixel, dark-current rate $d$ per pixel, $M$ reads with rms read noise $r$ per pixel, and an independently estimated mean background per pixel of variance ${\sigma }_{\hat{B}}^2$, an ideal aperture-sum budget isThe last term represents the propagated calibration uncertainty if it can be treated as a random variance; systematic residuals can instead bias the measurement and need an explicit error model. For a background estimated from $n_b$ independent, equally exposed sky pixels, one approximation is ${\sigma }_{\hat{B}}^2=[(b+d)t+M{r}^2]/n_b$. Correlated resampling, detector cross-talk or a spatially structured sky invalidates the independent-pixel simplification.

Photon counting and sky. Source photon shot noise has variance $S$ and cannot be calibrated away. Longer integration or greater throughput improves its fractional error as $S^{-1/2}$. Sky photons, including moonlight, airglow and scattered light, add their own Poisson fluctuations. A dark site, appropriate observing time and filter, good baffling and a smaller or optimally weighted source aperture reduce sky contributions. Better seeing or adaptive optics can concentrate a point source into fewer noise-bearing pixels. Sky subtraction removes the mean background but not the random sky photons already recorded, and the sky estimate adds uncertainty of its own.

Dark and read noise. Thermal dark current contributes a mean and shot noise. Cooling reduces its rate; well-matched dark frames remove a reproducible mean and map hot pixels, but do not undo the science exposure's random dark-current fluctuations. Each read adds electronic noise. At fixed total integration, many short exposures add more read variance than fewer long exposures, so use exposures long enough to become sky limited when saturation, cosmic rays and temporal changes permit. Multiple non-destructive reads can help on detectors that support them. Subtracting a bias frame corrects an electronic offset, not the stochastic read noise.

Calibration and non-Poisson effects. Pixel sensitivity differences, illumination gradients, fringes, nonlinearity, imperfect charge transfer and electronic pattern noise leave calibration errors. High-count flat fields, stable operating conditions, linearity calibration, dithering and appropriate detector models help, while the finite signal-to-noise of calibration frames propagates into the result. Cosmic-ray hits, hot pixels, satellite trails and scattered-light artefacts are not ordinary stationary Gaussian noise: identify or mask them, often using several exposures. Overaggressive clipping can remove real transient or compact-source signal.

Atmospheric scintillation adds correlated intensity fluctuations, especially in short bright-star measurements; seeing changes and guiding errors move flux between pixels or apertures. Differential photometry using appropriate reference stars and stable apertures reduces common variations but does not guarantee removal of chromatic or position-dependent effects. For a constant flux-limited source in independent stationary noise the final signal-to-noise usually grows as $\sqrt{t}$ once background limited; a systematic floor or correlated sky can stop that improvement. Distinguish improvement of calibration, which removes reproducible bias, from elimination of intrinsically random photon arrivals, which is impossible.