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Photometry and Spectroscopy
0.1 Photometry
Galaxy photometry is the quantitative measurement of light emitted by galaxies across different regions and wavelengths. Unlike stars, which appear as points, galaxies are extended objects whose images are blurred by atmospheric turbulence, a phenomenon known as seeing. Because of this, ground-based optical telescopes generally cannot distinguish details smaller than about $1/3''$.
Surface Brightness and Isophotes: The fundamental measurement in galaxy photometry is surface brightness ($I(\mathbf{x})$), defined as the amount of light per square arcsecond at a specific point in a galaxy’s image. Mathematically, it is expressed as: $$I(\mathbf{x}) = \frac{L}{4\pi D^2}$$ where $L$ is luminosity and $D$ is the physical diameter of a patch of the galaxy. Surface brightness is typically measured in mag arcsec$^{-2}$ or $L_\odot \text{ pc}^{-2}$. A crucial property of surface brightness is that it is independent of the observer’s distance, except at very large cosmological distances where the expansion of the Universe causes it to dim. Contours of constant surface brightness are called isophotes.
Structural Profiles: Astronomers use mathematical models to describe how a galaxy’s light is distributed:
Galactic Disks: The surface brightness of spiral and S0 disks generally follows an exponential profile: $I(R) = I(0) \exp(-R/h_R)$, where $h_R$ is the radial scale length (typically 1–10 kpc).
Bulges and Ellipticals: These systems are often modeled using the Sérsic formula: $I(R) = I(0) \exp[-(R/R_0)^{1/n}]$. A specific version of this where $n=4$ is known as the de Vaucouleurs $R^{1/4}$ law, which provides a good description for luminous elliptical galaxies.
Effective Radius ($R_e$): A standard measure of size, $R_e$ is the radius of a circle on the sky that encloses half of a galaxy’s total light.
Observational Challenges
Sky Brightness: The night sky itself emits light (from airglow and moonlight), which is often brighter than the faint outer parts of galaxies. Accurate photometry requires precise sky subtraction to isolate the galaxy’s light.
Detectors: Modern photometry primarily uses Charge-Coupled Devices (CCDs) and/or Complementary Metal-Oxide Semiconductor (CMOS), which have quantum efficiencies above 90% in the visible spectrum. CCD/CMOS images must be corrected using flat fields to account for pixel-to-pixel sensitivity variations and calibrated using stars of known brightness.
Dust Extinction: Interstellar dust in both the target galaxy and our own Milky Way absorbs and scatters light, necessitating corrections for reddening and dimming.
Cosmological Effects: For distant galaxies at high redshift ($z$), several factors complicate photometry:
Surface Brightness Dimming: The bolometric surface brightness decreases rapidly as $(1+z)^{-4}$, making high-redshift systems extremely difficult to observe.
K-Correction: Because the expansion of space shifts a galaxy’s spectrum to longer wavelengths, a fixed observational filter samples different parts of the galaxy’s rest-frame light depending on its redshift.
Photometric Redshifts: By comparing a galaxy’s apparent brightness across multiple filter bands (e.g., U, B, V, R, I), astronomers can estimate its redshift without taking a full spectrum.