Part IV: Frontiers | Chapter 4

Observational Techniques

The tools of modern astrophysics: telescope design, detector physics, spectroscopy, interferometry, and astronomical survey strategies

Overview

Astrophysics is fundamentally an observational science. Our understanding of the Universe depends entirely on the photons, particles, and gravitational waves we can collect and analyze. Modern astronomical instrumentation spans the electromagnetic spectrum from radio waves ($\lambda \sim 1$ m) to gamma rays ($\lambda \sim 10^{-15}$ m), with each wavelength regime requiring different technologies and revealing different physical processes.

In this chapter we derive the fundamental limits on telescope performance (diffraction, sensitivity), explain detector physics from CCDs to bolometers, develop the theory of astronomical spectroscopy and the information extractable from spectra, derive the principles of interferometry, and discuss survey design optimization.

1. Telescope Optics and the Diffraction Limit

The angular resolution and light-gathering power of a telescope are determined by its aperture diameter $D$.

1.1 The Rayleigh Criterion

The diffraction pattern of a circular aperture (the Airy pattern) has its first minimum at an angular radius:

$$\boxed{\theta_{\text{diff}} = 1.22\frac{\lambda}{D} \approx 0.025''\left(\frac{\lambda}{500\,\text{nm}}\right)\left(\frac{D}{10\,\text{m}}\right)^{-1}}$$

Two point sources are considered resolved when their angular separation exceeds $\theta_{\text{diff}}$ (the Rayleigh criterion). The 39-m ELT will achieve $\theta_{\text{diff}} \approx 0.003''$ at visible wavelengths, but ground-based telescopes are limited by atmospheric turbulence (seeing).

1.2 Atmospheric Seeing

Turbulence in the Earth's atmosphere distorts the incoming wavefront, degrading the angular resolution to the seeing limit:

$$\theta_{\text{seeing}} \approx 0.98\frac{\lambda}{r_0}$$

where $r_0$ is the Fried parameter (the coherence length of the atmosphere), typically $r_0 \approx 10\text{--}20$ cm at visible wavelengths at good sites. This gives $\theta_{\text{seeing}} \approx 0.5''\text{--}1.0''$, independent of telescope aperture for $D > r_0$.

1.3 Adaptive Optics

Adaptive optics (AO) systems correct for atmospheric turbulence in real-time using a deformable mirror controlled by wavefront sensors. The number of actuators needed scales as:

$$N_{\text{act}} \approx \left(\frac{D}{r_0}\right)^2$$

For a 10-m telescope at visible wavelengths, $N_{\text{act}} \sim 2500$. AO systems use either natural guide stars or laser guide stars (sodium layer beacons at $\sim 90$ km altitude) as wavefront references. The Strehl ratio (ratio of peak intensity to the diffraction-limited peak) measures AO performance; modern systems achieve Strehl $> 90\%$ in the near-infrared.

2. Detector Physics

The quantum efficiency and noise properties of detectors determine the ultimate sensitivity of astronomical observations.

2.1 CCD Signal-to-Noise Ratio

For a charge-coupled device (CCD) observing a source with photon rate $S$(photons s$^{-1}$ pixel$^{-1}$), the signal-to-noise ratio after integration time $t$ is:

$$\boxed{\text{SNR} = \frac{S\,t}{\sqrt{S\,t + n_{\text{pix}}(B\,t + D\,t + R^2)}}}$$

where $B$ is the sky background rate per pixel, $D$ is the dark current, $R$ is the read noise (in electrons), and $n_{\text{pix}}$is the number of pixels in the extraction aperture. Two regimes:

Source-limited: $\text{SNR} \propto \sqrt{t}$(bright sources)

Background-limited: $\text{SNR} \propto \sqrt{t/n_{\text{pix}}}$(faint sources in high background)

2.2 The Photon Collecting Power

The number of photons collected from a source of magnitude $m$ in time $t$with a telescope of diameter $D$ and quantum efficiency $\eta$ is:

$$N_\gamma = F_0\,10^{-m/2.5}\,\frac{\pi D^2}{4}\,\eta\,\frac{\Delta\lambda}{\lambda}\,t$$

where $F_0$ is the zero-point flux. Modern CCDs achieve $\eta > 90\%$at peak wavelength. The JWST's 6.5-m primary mirror and low-background space environment enable detection of sources $\sim 100$ times fainter than the largest ground-based telescopes.

3. Astronomical Spectroscopy

Spectroscopy is the most powerful diagnostic tool in astrophysics, revealing composition, temperature, density, velocity, and magnetic fields from the analysis of spectral lines.

3.1 Spectral Resolution

The resolving power of a spectrograph is defined as:

$$\boxed{R = \frac{\lambda}{\Delta\lambda} = \frac{c}{\Delta v}}$$

where $\Delta\lambda$ is the minimum resolvable wavelength difference. Different science goals require different resolutions:

$R \sim 100$: Broadband photometry and spectral energy distributions

$R \sim 1000\text{--}5000$: Galaxy redshift surveys, emission line identification

$R \sim 20{,}000\text{--}50{,}000$: Stellar abundance analysis, ISM kinematics

$R > 100{,}000$: Exoplanet radial velocities, precise isotope ratios

3.2 Information Content of Spectral Lines

Each spectral line encodes multiple physical parameters:

Line center ($\lambda_0$): Identifies the atomic/molecular transition; Doppler shift gives radial velocity via $\Delta\lambda/\lambda_0 = v_r/c$.

Line width: Thermal broadening gives $\Delta\lambda_{\text{th}} = (\lambda_0/c)\sqrt{2k_BT/m}$; turbulent broadening reflects gas dynamics; pressure broadening probes density.

Line strength (equivalent width): Related to column density and abundance via the curve of growth:

$$W_\lambda = \int\left(1 - \frac{F_\lambda}{F_c}\right)d\lambda$$

The curve of growth relates $W_\lambda$ to the column density $N$through three regimes: linear (optically thin), flat (saturated Doppler core), and square-root (damping wings dominate).

4. Interferometry

Interferometry combines signals from multiple telescopes to achieve angular resolution equivalent to a telescope with diameter equal to the maximum baseline.

4.1 The Van Cittert-Zernike Theorem

The fundamental theorem of radio interferometry states that the complex visibility measured by a baseline $\mathbf{b}$ is the Fourier transform of the sky brightness distribution:

$$\boxed{V(\mathbf{u}) = \int I(\mathbf{l})\,e^{-2\pi i\,\mathbf{u}\cdot\mathbf{l}}\,d^2l}$$

where $\mathbf{u} = \mathbf{b}/\lambda$ is the baseline in wavelength units and $\mathbf{l}$ is the direction cosine on the sky. Each baseline samples one point in the $(u,v)$ plane. Image reconstruction involves inverting this Fourier relationship, accounting for incomplete $(u,v)$ coverage.

4.2 Resolution and Sensitivity

The angular resolution of an interferometer is:

$$\theta_{\text{int}} = \frac{\lambda}{B_{\text{max}}}$$

ALMA (baseline up to 16 km at $\lambda = 1$ mm) achieves $\theta \sim 15$ mas. The EHT (Earth-diameter baselines at$\lambda = 1.3$ mm) achieves $\theta \sim 20\,\mu$as, sufficient to resolve the shadow of the M87 supermassive black hole.

4.3 Aperture Synthesis

As the Earth rotates, each baseline traces an ellipse in the $(u,v)$ plane, progressively filling in Fourier coverage. This technique (aperture synthesis), developed by Martin Ryle (Nobel Prize 1974), allows a small number of antennas to synthesize a much larger effective aperture. The DIRTY image (inverse Fourier transform of the sampled visibilities) is deconvolved using algorithms such as CLEAN (Hogbom 1974) to produce the final image.

5. Astronomical Survey Design

Modern astrophysics increasingly relies on large-area surveys that observe millions to billions of objects.

5.1 The Survey Figure of Merit

For a survey with telescope aperture $D$, field of view $\Omega$, and total time $T$, the product:

$$\boxed{A\Omega = \frac{\pi D^2}{4}\,\Omega}$$

is the etendue, the fundamental figure of merit for survey speed. The Vera C. Rubin Observatory (8.4 m primary, 9.6 deg$^2$ field) has the highest etendue of any optical telescope, enabling the Legacy Survey of Space and Time (LSST) to image the entire visible sky every 3 nights.

5.2 Depth vs Area Tradeoff

For a fixed total time $T$, the limiting magnitude scales as:

$$m_{\text{lim}} \propto 2.5\log\left(\frac{T}{A_{\text{survey}}}\right) + \text{const}$$

Deeper surveys cover smaller areas. The optimal strategy depends on the science goal: rare objects (e.g., high-z quasars) require wide area; faint objects (e.g., distant galaxy populations) require depth. The SDSS (wide/shallow), GOODS (deep/narrow), and LSST (wide/moderate depth with time domain) represent different optimization choices.

5.3 Multi-Object Spectroscopy

Modern spectrographs use fiber-fed multi-object systems to observe hundreds to thousands of objects simultaneously. DESI deploys 5000 robotically-positioned fibers across a 3-degree field, enabling spectroscopic redshifts of 40 million galaxies over 5 years. The multiplex advantage scales linearly with the number of fibers, making survey efficiency proportional to the product of etendue and multiplex factor.

Applications

Next-Generation Facilities

The coming decade will see transformative new facilities: the Extremely Large Telescope (ELT, 39 m), the Giant Magellan Telescope (GMT, 24.5 m), the Thirty Meter Telescope (TMT, 30 m), the Roman Space Telescope (2.4 m with wide-field infrared imaging), the Vera Rubin Observatory (LSST), the Square Kilometre Array (SKA) in radio, and the Laser Interferometer Space Antenna (LISA) for millihertz gravitational waves. These instruments will probe the earliest galaxies, characterize exoplanet atmospheres, map dark energy, and detect gravitational waves from supermassive black hole mergers.

Machine Learning in Astronomy

The exponential growth in data volume (LSST will produce $\sim 20$ TB per night) necessitates automated analysis. Machine learning techniques are used for photometric redshift estimation, galaxy morphological classification, transient detection and classification, spectral analysis, and gravitational lens discovery. Deep learning models trained on simulations and existing surveys will be essential for extracting science from next-generation datasets.

Historical Notes

Galileo's telescope (1609) launched observational astronomy with a $\sim 4$ cm aperture. Newton invented the reflecting telescope in 1668. William Herschel discovered infrared radiation in 1800 by measuring temperature beyond the red end of the solar spectrum. Photography replaced visual observation in the late 19th century, with the Harvard spectral classification (Annie Jump Cannon, 1901–1924) establishing the OBAFGKM sequence. CCDs revolutionized astronomy in the 1970s–1980s, replacing photographic plates with linear, high-quantum-efficiency digital detectors. Karl Jansky's discovery of cosmic radio emission in 1932 opened the first non-optical window. Space-based observatories (Hubble, 1990; Chandra, 1999; Spitzer, 2003; JWST, 2021) overcome atmospheric absorption and achieve diffraction-limited performance across the electromagnetic spectrum.

The evolution from visual observation through photographic plates to modern digital detectors has increased the sensitivity of astronomical measurements by roughly a factor of $10^{10}$ over the past four centuries. Each technological advance has revealed new phenomena: radio astronomy discovered pulsars, quasars, and the CMB; X-ray astronomy revealed accreting black holes and hot gas in galaxy clusters; infrared astronomy pierced the dust obscuring star-forming regions and the Galactic center; gamma-ray astronomy detected the most energetic processes in the Universe. The multi-wavelength approach — combining data across the electromagnetic spectrum — is now standard practice, with facilities coordinating through rapid alert systems (GCN, TNS, ATEL) to enable prompt follow-up of transient events.

The Multi-Wavelength Universe

Different wavelengths probe different physical processes and environments. Radio emission traces synchrotron radiation from relativistic electrons, thermal free-free emission from HII regions, and molecular line emission (CO, HCN) from cold gas. Infrared traces dust-reprocessed starlight and the redshifted UV emission from high-$z$ galaxies. Optical and UV trace direct starlight, emission lines from ionized gas, and absorption features from the interstellar and intergalactic medium. X-rays trace hot gas ($T > 10^6$ K) in clusters, accretion onto compact objects, and coronal emission from active stars. Gamma-rays trace the most extreme particle acceleration in jets, magnetars, and cosmic ray interactions.

Photometry and Calibration

Accurate photometric calibration is essential for all quantitative astrophysics. The AB magnitude system defines magnitudes in terms of specific flux density:

$$m_{\text{AB}} = -2.5\log_{10}\left(\frac{f_\nu}{3631\,\text{Jy}}\right)$$

Atmospheric extinction removes photons with a wavelength-dependent optical depth that increases with airmass $X = \sec z$ (where $z$ is the zenith angle):

$$m_{\text{obs}} = m_{\text{true}} + k(\lambda)\,X$$

where $k(\lambda)$ is the extinction coefficient, typically$\sim 0.1\text{--}0.3$ mag/airmass at visible wavelengths at good sites. Observations of photometric standard stars at different airmasses determine $k(\lambda)$and the instrumental zero point. Modern surveys use overlapping observations and "ubercalibration" techniques to achieve photometric uniformity of $\sim 1\text{--}2\%$ across the survey footprint. Sub-percent photometric accuracy is required for precision cosmology applications such as Type Ia supernova distance measurements and photometric redshift estimation.

Computational Exploration

The following simulation computes the diffraction limit and Airy pattern, generates a CCD signal-to-noise calculator, models spectral line profiles with different broadening mechanisms, and illustrates interferometric (u,v) coverage.

Diffraction Patterns, SNR Calculator, Spectral Lines, and Interferometry

Python

script.py280 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0a0a0a')
for ax in axes.flat:
    ax.set_facecolor('#0f0f0f')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Plot 1: Airy pattern and diffraction limit ---
ax1 = axes[0, 0]

def airy_pattern(theta, D, lam):
    """Normalized Airy pattern intensity"""
    x = np.pi * D * theta / lam
    # Handle x=0
    result = np.ones_like(x, dtype=float)
    mask = x != 0
    # J1(x)/x using series expansion for accuracy
    # For simplicity, use the formula: (2J1(x)/x)^2
    # J1(x) ~ x/2 - x^3/16 + x^5/384 - ...
    x_m = x[mask]
    # Compute J1 using simple series
    j1 = np.zeros_like(x_m)
    for n in range(20):
        sign = (-1)**n
        j1 += sign * (x_m/2)**(2*n+1) / (np.math.factorial(n) * np.math.factorial(n+1))
    result[mask] = (2 * j1 / x_m)**2
    return result

theta_arcsec = np.linspace(-0.1, 0.1, 1000)
theta_rad = theta_arcsec * np.pi / (180 * 3600)

lam = 500e-9  # 500 nm

telescopes = [
    ('HST (2.4 m)', 2.4, '#44bbff'),
    ('Keck (10 m)', 10.0, '#ffa500'),
    ('ELT (39 m)', 39.0, '#ff4444'),
]

for label, D, col in telescopes:
    I = airy_pattern(theta_rad, D, lam)
    # Rayleigh limit
    theta_R = 1.22 * lam / D * 180 * 3600 / np.pi
    ax1.plot(theta_arcsec * 1000, I, color=col, linewidth=2.5, label=f'{label}: {theta_R*1000:.1f} mas')
    ax1.axvline(x=theta_R*1000, color=col, linestyle=':', linewidth=1, alpha=0.5)

# Seeing limit
ax1.axvspan(-500, -250, alpha=0, color='white')
ax1.axvline(x=500/2, color='#888', linestyle='--', linewidth=1.5, alpha=0.5)
ax1.text(55, 0.8, 'Typical seeing\n(0.5")', color='#888', fontsize=9, ha='center')

ax1.set_xlabel('Angular offset (milliarcsec)', fontsize=12)
ax1.set_ylabel('Normalized Intensity', fontsize=12)
ax1.set_title('Airy Diffraction Patterns (500 nm)', fontsize=14, fontweight='bold')
ax1.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax1.set_xlim(-100, 100)
ax1.set_ylim(0, 1.05)
ax1.grid(True, alpha=0.15)

# --- Plot 2: CCD SNR vs magnitude ---
ax2 = axes[0, 1]

mag_range = np.linspace(15, 30, 200)

# Photon rate from magnitude (V-band, approximate)
# For V=0, ~1000 photons/s/cm^2/Angstrom
# Through a filter with delta_lambda = 1000 A

def ccd_snr(mag, D_m, t_sec, sky_mag=21.5, seeing=0.8, eta=0.8, R_noise=5, dark=0.01):
    """CCD signal-to-noise ratio"""
    # Zero point: ~1e10 photons/s/m^2 for V=0 through broadband filter
    F_0 = 1e10  # photons/s/m^2 for mag=0
    S = F_0 * 10**(-mag/2.5) * np.pi * (D_m/2)**2 * eta  # signal rate (photons/s)
    B_sky = F_0 * 10**(-sky_mag/2.5) * np.pi * (D_m/2)**2 * eta  # sky per arcsec^2

n_pix = np.pi * (seeing/0.2)**2  # pixels (0.2"/pix plate scale)
    B_total = B_sky * seeing**2  # sky per extraction aperture

signal = S * t_sec
    noise = np.sqrt(S * t_sec + n_pix * (B_total * t_sec + dark * t_sec + R_noise**2))
    return signal / noise

# Different telescopes and exposure times
configs = [
    ('2m, 1hr', 2.0, 3600, '#44bbff'),
    ('8m, 1hr', 8.0, 3600, '#ffa500'),
    ('8m, 10hr', 8.0, 36000, '#ff4444'),
    ('JWST 6.5m, 10hr (space)', 6.5, 36000, '#ffdd00'),
]

for label, D, t, col in configs:
    sky = 21.5 if 'JWST' not in label else 27.0  # much darker in space
    snr = ccd_snr(mag_range, D, t, sky_mag=sky)
    ax2.semilogy(mag_range, snr, color=col, linewidth=2.5, label=label)

# Detection thresholds
ax2.axhline(y=5, color='#ff4444', linestyle='--', linewidth=1.5, alpha=0.5, label='5-sigma detection')
ax2.axhline(y=10, color='#888', linestyle=':', linewidth=1, alpha=0.5)

ax2.set_xlabel('Apparent Magnitude (V)', fontsize=12)
ax2.set_ylabel('Signal-to-Noise Ratio', fontsize=12)
ax2.set_title('CCD SNR vs Source Brightness', fontsize=14, fontweight='bold')
ax2.legend(fontsize=8, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax2.set_xlim(15, 30)
ax2.set_ylim(0.1, 1e4)
ax2.grid(True, alpha=0.15)

# --- Plot 3: Spectral line profiles ---
ax3 = axes[1, 0]

# Central wavelength: H-alpha = 6563 Angstroms
lam_0 = 6563.0  # Angstroms
lam_range = np.linspace(6555, 6571, 1000)

# Natural broadening (Lorentzian)
gamma_nat = 0.01  # Angstroms (very narrow)
L_nat = (gamma_nat / (2*np.pi)) / ((lam_range - lam_0)**2 + (gamma_nat/2)**2)
L_nat /= max(L_nat)

# Thermal broadening (Gaussian) - T = 10000 K, hydrogen
k_B = 1.381e-23
m_H = 1.673e-27
T = 10000
v_th = np.sqrt(2 * k_B * T / m_H)  # m/s
sigma_th = lam_0 * v_th / 3e8  # Angstroms
G_th = np.exp(-0.5 * ((lam_range - lam_0) / sigma_th)**2)

# Turbulent broadening (v_turb = 20 km/s)
v_turb = 20e3  # m/s
sigma_turb = lam_0 * v_turb / 3e8
G_turb = np.exp(-0.5 * ((lam_range - lam_0) / sigma_turb)**2)

# Combined Voigt (thermal + natural: simplified as convolution approximation)
sigma_total = np.sqrt(sigma_th**2 + sigma_turb**2)
G_combined = np.exp(-0.5 * ((lam_range - lam_0) / sigma_total)**2)

# Pressure broadened (Lorentzian with larger width)
gamma_press = 0.5  # Angstroms
L_press = (gamma_press / (2*np.pi)) / ((lam_range - lam_0)**2 + (gamma_press/2)**2)
L_press /= max(L_press)

ax3.plot(lam_range, G_th, color='#44bbff', linewidth=2.5, label=f'Thermal (T={T} K)')
ax3.plot(lam_range, G_turb, '--', color='#ffa500', linewidth=2, label=f'Turbulent (v=20 km/s)')
ax3.plot(lam_range, G_combined, color='#ff4444', linewidth=2.5, label='Combined Gaussian')
ax3.plot(lam_range, L_press, ':', color='#ffdd00', linewidth=2, label='Pressure broadened')

# Redshifted line
z = 0.001  # v = 300 km/s
lam_shifted = lam_0 * (1 + z)
ax3.axvline(x=lam_shifted, color='#ff69b4', linestyle='--', linewidth=1.5, alpha=0.5)
ax3.text(lam_shifted + 0.2, 0.9, f'z={z}\n(v=300 km/s)', color='#ff69b4', fontsize=9)

ax3.set_xlabel('Wavelength (Angstroms)', fontsize=12)
ax3.set_ylabel('Normalized Flux', fontsize=12)
ax3.set_title('H-alpha Line Broadening Mechanisms', fontsize=14, fontweight='bold')
ax3.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax3.set_xlim(6555, 6571)
ax3.set_ylim(0, 1.1)
ax3.grid(True, alpha=0.15)

# --- Plot 4: Interferometric (u,v) coverage ---
ax4 = axes[1, 1]

# Simulate VLA-like array (Y-shaped, 27 antennas)
N_ant = 27
arm_length = 21e3  # meters (21 km max)

# Generate antenna positions (Y-shape)
angles_arms = [0, 120, 240]  # degrees
ant_x = []
ant_y = []

for angle in angles_arms:
    n_per_arm = N_ant // 3
    distances = np.logspace(1, np.log10(arm_length), n_per_arm)
    for d in distances:
        ant_x.append(d * np.cos(angle * np.pi / 180))
        ant_y.append(d * np.sin(angle * np.pi / 180))

ant_x = np.array(ant_x)
ant_y = np.array(ant_y)

# Compute baselines
lam_obs = 0.06  # 6 cm = 5 GHz
u_all = []
v_all = []

# At different hour angles (Earth rotation synthesis)
for ha in np.linspace(-6, 6, 24):  # hours
    ha_rad = ha * np.pi / 12
    dec = 30 * np.pi / 180  # source declination

for i in range(len(ant_x)):
        for j in range(i+1, len(ant_x)):
            bx = ant_x[j] - ant_x[i]
            by = ant_y[j] - ant_y[i]

# Rotate by hour angle (simplified projection)
            u = (bx * np.sin(ha_rad) + by * np.cos(ha_rad)) / lam_obs
            v = (-bx * np.sin(dec)*np.cos(ha_rad) + by * np.sin(dec)*np.sin(ha_rad)) / lam_obs

u_all.append(u)
            v_all.append(v)
            u_all.append(-u)  # conjugate
            v_all.append(-v)

u_all = np.array(u_all) / 1e6  # mega-lambda
v_all = np.array(v_all) / 1e6

# Subsample for plotting
idx = np.random.default_rng(42).choice(len(u_all), min(50000, len(u_all)), replace=False)
ax4.scatter(u_all[idx], v_all[idx], s=0.2, color='#ffa500', alpha=0.3)

ax4.set_xlabel('u (Mega-lambda)', fontsize=12)
ax4.set_ylabel('v (Mega-lambda)', fontsize=12)
ax4.set_title('VLA-like (u,v) Coverage (12hr synthesis)', fontsize=14, fontweight='bold')
ax4.set_aspect('equal')
max_uv = max(abs(u_all[idx]).max(), abs(v_all[idx]).max()) * 1.1
ax4.set_xlim(-max_uv, max_uv)
ax4.set_ylim(-max_uv, max_uv)
ax4.grid(True, alpha=0.15)

# Annotate resolution
B_max = arm_length * 2
theta_res = lam_obs / B_max * 180 * 3600 / np.pi  # arcsec
ax4.text(0.05, 0.95, f'Resolution: {theta_res:.2f}"\nat 5 GHz', transform=ax4.transAxes,
         color='#ffdd00', fontsize=11, verticalalignment='top')

# Print summary
print("=" * 65)
print("OBSERVATIONAL TECHNIQUES: KEY RESULTS")
print("=" * 65)

print("\nDiffraction limits at 500 nm:")
for label, D, _ in telescopes:
    theta_R = 1.22 * 500e-9 / D * 180 * 3600 / np.pi * 1000
    print(f"  {label}: theta = {theta_R:.1f} mas")

print(f"\nTypical seeing: 0.5-1.0 arcsec")
print(f"  Fried parameter r_0 ~ 10-20 cm at visible wavelengths")

print("\n5-sigma detection limits (V-band, 1 hour):")
for label, D, t, _ in configs:
    sky = 21.5 if 'JWST' not in label else 27.0
    for m in range(20, 32):
        snr = ccd_snr(m, D, t, sky_mag=sky)
        if snr < 5:
            print(f"  {label}: m_lim ~ {m-1}")
            break

print(f"\nSpectral line widths (H-alpha at T=10000K):")
print(f"  Thermal: sigma = {sigma_th:.3f} A ({v_th/1e3:.1f} km/s)")
print(f"  Turbulent: sigma = {sigma_turb:.3f} A (20 km/s)")
print(f"  Combined: sigma = {sigma_total:.3f} A")

print(f"\nInterferometric resolution:")
print(f"  VLA at 5 GHz (B=42 km): {theta_res:.2f} arcsec")
print(f"  ALMA at 345 GHz (B=16 km): {0.001/16e3 * 206265:.3f} arcsec")
print(f"  EHT at 230 GHz (B=10000 km): {0.0013/10e6 * 206265e6:.0f} micro-arcsec")

print(f"\nSurvey etendue (A*Omega):")
surveys = [("SDSS (2.5m, 3 deg^2)", 2.5, 3), ("Rubin/LSST (8.4m, 9.6 deg^2)", 8.4, 9.6),
           ("Roman (2.4m, 0.28 deg^2)", 2.4, 0.28)]
for name, D, omega in surveys:
    AO = np.pi * (D/2)**2 * omega
    print(f"  {name}: AO = {AO:.1f} m^2 deg^2")

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
print("\nPlot saved to output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Chapter Summary

Telescope resolution is fundamentally limited by diffraction ($\theta = 1.22\lambda/D$) but practically limited by atmospheric seeing ($\sim 0.5''\text{--}1''$) for ground-based telescopes. Adaptive optics recovers diffraction-limited performance by correcting wavefront distortions.

CCD signal-to-noise follows Poisson statistics with contributions from source photons, sky background, dark current, and read noise. Spectroscopy at resolutions $R = \lambda/\Delta\lambda$ from $\sim 100$to $> 100{,}000$ reveals radial velocities, temperatures, abundances, and gas dynamics.

Interferometry achieves angular resolutions far beyond single telescopes through the Van Cittert-Zernike theorem relating visibilities to the sky brightness distribution. Survey design optimizes the etendue $A\Omega$ for the science goal, balancing area against depth.

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